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First of all Bishul Akum simply means Food cooked by a gentile. All the Halachos are stated in Shulchan aruch Yore dea 113 and though intermarriage may be one reason for this prohibition, no where does it state fulfilling Bishul akum is dependent on likely-hood of intermarriage as Mitzvos are not subjective unless specified.
Indeed the Shach (with consesus of Taz, Beer heitev and Acharonim)ibid quotes the Ramban in Seif 4 that the Halachic consensus is that we don't differentiate between cooking of regular Gentiles those that are cooking as a job where they are not interested in intermarriage:

תשובת הרמב"ן סי' קמ"ט דיש מרבותינו מתירין בשפחות מטעם דאין איסור משום בישולי עובד כוכבים אלא בעושה מרצונו משום גזירת חתנות אבל אלו השפחות והעבדים שלנו שעושים בע"כ בין ירצו בין לא ירצו אין בזה קירוב הדעת ומ"מ אין דבריהם מחוורין ואין אנו סומכין ע"ז ואנו נוהגין איסור אפילו בדיעבד

Secondly a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother. Rambam Hilchot Melachim 9,5:

שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה

This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother). Indeed Tamar wanted to marry Amnon her brother through her father Shmuel 2 13,13 she was conceived when her mother was not Jewish yet (see Sanhedrin 21a from here we learn that Yichud is prohibitted)

Thirdly this sister probably hangs out with gentile friends so even if she feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.

First of all Bishul Akum Food cooked by a gentile applies even if there is no worry of intermarriage Rema Yore Dea 112,1:

אסרו חכמים לאכול פת של עובדי כוכבים, משום חתנות. (ואפילו במקום דליכא משום חתנות, אסור) (רשב"א סימן רמ"ח)

Secondly, since a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother there is concern of marriage in this situation. Rambam Hilchot Melachim 9,5 says:
שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה
This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother). Indeed Tamar wanted to marry Amnon her brother through her father Shmuel 2 13,13 she was conceived when her mother was not Jewish yet (see Sanhedrin 21a from here we learn that Yichud is prohibitted)

Thirdly in this situation, this sister probably hangs out with gentile friends so even if she personally feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.

First of all Bishul Akum simply means Food cooked by a gentile. All the Halachos are stated in Shulchan aruch Yore dea 113 and though intermarriage may be one reason for this prohibition, no where does it state fulfilling Bishul akum is dependent on likely-hood of intermarriage as Mitzvos are not subjective unless specified.
Indeed the Shach (with consesus of Taz, Beer heitev and Acharonim)ibid quotes the Ramban in Seif 4 that the Halachic consensus is that we don't differentiate between cooking of regular Gentiles those that are cooking as a job where they are not interested in intermarriage:

תשובת הרמב"ן סי' קמ"ט דיש מרבותינו מתירין בשפחות מטעם דאין איסור משום בישולי עובד כוכבים אלא בעושה מרצונו משום גזירת חתנות אבל אלו השפחות והעבדים שלנו שעושים בע"כ בין ירצו בין לא ירצו אין בזה קירוב הדעת ומ"מ אין דבריהם מחוורין ואין אנו סומכין ע"ז ואנו נוהגין איסור אפילו בדיעבד

Secondly a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother. Rambam Hilchot Melachim 9,5:

שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה

This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother).

Thirdly this sister probably hangs out with gentile friends so even if she feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.

First of all Bishul Akum simply means Food cooked by a gentile. All the Halachos are stated in Shulchan aruch Yore dea 113 and though intermarriage may be one reason for this prohibition, no where does it state fulfilling Bishul akum is dependent on likely-hood of intermarriage as Mitzvos are not subjective unless specified.
Indeed the Shach (with consesus of Taz, Beer heitev and Acharonim)ibid quotes the Ramban in Seif 4 that the Halachic consensus is that we don't differentiate between cooking of regular Gentiles those that are cooking as a job where they are not interested in intermarriage:

תשובת הרמב"ן סי' קמ"ט דיש מרבותינו מתירין בשפחות מטעם דאין איסור משום בישולי עובד כוכבים אלא בעושה מרצונו משום גזירת חתנות אבל אלו השפחות והעבדים שלנו שעושים בע"כ בין ירצו בין לא ירצו אין בזה קירוב הדעת ומ"מ אין דבריהם מחוורין ואין אנו סומכין ע"ז ואנו נוהגין איסור אפילו בדיעבד

Secondly a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother. Rambam Hilchot Melachim 9,5:

שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה

This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother). Indeed Tamar wanted to marry Amnon her brother through her father Shmuel 2 13,13 she was conceived when her mother was not Jewish yet (see Sanhedrin 21a from here we learn that Yichud is prohibitted)

Thirdly this sister probably hangs out with gentile friends so even if she feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.

First of all Bishul Akum simply means Food cooked by a gentile. All the Halachos are stated in Shulchan aruch Yore dea 113 and though intermarriage may be one reason for this prohibition, no where does it state fulfilling Bishul akum is dependent on likely-hood of intermarriage as Mitzvos are not subjective unless specified.
Indeed the Shach ibid quotes the Ramban in Seif 4 that the Halachic consensus is that we don't differentiate between cooking of regular Gentiles those that are cooking as a job where they are not interested in intermarriage:

תשובת הרמב"ן סי' קמ"ט דיש מרבותינו מתירין בשפחות מטעם דאין איסור משום בישולי עובד כוכבים אלא בעושה מרצונו משום גזירת חתנות אבל אלו השפחות והעבדים שלנו שעושים בע"כ בין ירצו בין לא ירצו אין בזה קירוב הדעת ומ"מ אין דבריהם מחוורין ואין אנו סומכין ע"ז ואנו נוהגין איסור אפילו בדיעבד

Secondly a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother. Rambam Hilchot Melachim 9,5:

שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה

This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother).

Thirdly this sister probably hangs out with gentile friends so even if she feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.

First of all Bishul Akum simply means Food cooked by a gentile. All the Halachos are stated in Shulchan aruch Yore dea 113 and though intermarriage may be one reason for this prohibition, no where does it state fulfilling Bishul akum is dependent on likely-hood of intermarriage as Mitzvos are not subjective unless specified.
Indeed the Shach (with consesus of Taz, Beer heitev and Acharonim)ibid quotes the Ramban in Seif 4 that the Halachic consensus is that we don't differentiate between cooking of regular Gentiles those that are cooking as a job where they are not interested in intermarriage:

תשובת הרמב"ן סי' קמ"ט דיש מרבותינו מתירין בשפחות מטעם דאין איסור משום בישולי עובד כוכבים אלא בעושה מרצונו משום גזירת חתנות אבל אלו השפחות והעבדים שלנו שעושים בע"כ בין ירצו בין לא ירצו אין בזה קירוב הדעת ומ"מ אין דבריהם מחוורין ואין אנו סומכין ע"ז ואנו נוהגין איסור אפילו בדיעבד

Secondly a Gentile according to Halacha is allowed to marry a sibling who shares a common father but not a common mother. Rambam Hilchot Melachim 9,5:

שש עריות אסורות על בני נח: האם ואשת האב ואשת איש ואחותו מאמו וזכור ובהמה

This means his sister from his father might want to marry him as they are not related according to Halacha of bnei Noach (it might not bother her that he is forbidden to her because he's Jewish and shes not) and if she converts they would be able to marry each other as stated in Shulchan Aruch 269,3 (though a convert cannot marry a sister from the mother).

Thirdly this sister probably hangs out with gentile friends so even if she feels repulsed to marry him he can still frequent her friends/relatives at meals which she cooks.