Assuming, as per the stipulation at the end of your question, that execution might be considered a reasonable punishment for the crimes committed, the question of whether or not one may be subject to a non-Jewish court's decision is a matter of debate.
In the early literature, non-Jewish courts (which were not "secular", strictly speaking) are referred to as arka'ot (ערכאות), although the precise etymology of this word is debated, and as agoriyot (אגוריות). There is a longstanding prohibition against having one's case heard in one, the oldest source for which is the Talmud Bavli:
ר"ט אומר כל מקום שאתה מוצא אגוריאות של עובדי כוכבים אע"פ שדיניהם כדיני
ישראל אי אתה רשאי להיזקק להם שנאמר ואלה המשפטים אשר תשים לפניהם לפניהם
ולא לפני עובדי כוכבים דבר אחר לפניהם ולא לפני הדיוטות
Rabbi Tarfon says, "Any place that you find non-Jewish courts
(agoriyot), even though their laws be the same as the laws of Israel,
you are not allowed to resort to them, since it says "these are the
laws that you shall set before them" (Exodus 21:1). Before them
and not before non-Jews. Alternatively, before them and not
before lay tribunals.
• Babylonian Talmud, Gittin 88b
This passage is quoted in the Sheiltot of R' Ahai Gaon (שאילתא ב', בראשית), although it adds the stipulation that this is even if the non-Jewish courts are operating in accordance with biblical law (אע"ג דקא דיינין דינא דאורייתא), and it gives the ruling in the name of Rabbi Meir instead of Rabbi Tarfon.
The same prohibition is to be found twice in Midrash Tanchuma: Mishpatim §3 (where it is presented anonymously) and §6 (where it is given in the name of Rabbi Shim'on, but in which it is otherwise identical to the passage in Gittin 88b).
If you consult the mediaeval mefarshim on Exodus 21:1, you will find the same ruling (quoted from the Tanchuma and from Gittin 88b) in the commentaries of both Rashi and Ramban. Rashi's wording is of particular interest:
לפניהם ולא לפני עובדי כוכבים ומזלות ואפילו ידעת בדין אחד שהם דנין אותו
כדיני ישראל אל תביאהו בערכאות שלהם שהמביא דיני ישראל לפני עובדי כוכבים
ומזלות מחלל את השם ומיקר את שם האלילים להשביחם
Before them, but not before non-Jews. And even if you know that they
are operating with the same law that the Jews employ, do not bring him
to their courts (arka'ot), for whoever brings Jewish cases before
non-Jews desecrates the name of God, and elevates foreign gods with
• Rashi, Exodus 21:1
As halakha, this ruling is to be found in the Rambam as follows:
כל הדן בדיני גוים ובערכאות שלהם אע"פ שהיו דיניהם כדיני ישראל הרי זה
רשע וכאלו חרף והרים יד בתורת משה רבינו שנ' ואלה המשפטים אשר תשים
לפניהם לפניהם ולא לפני גוים לפניהם ולא לפני הדיוטות
All those who try their cases in accordance with non-Jewish law or in
non-Jewish courts (arka'ot), even if their laws are as the laws of
Israel, is a wicked person and it is as though he insulted and struck
the Torah of Moses, since it says "These are the laws that you should
place before them". Before them, and not before non-Jews. Before
them, and not before lay tribunals.
- Rambam, Hilkhot Sanhedrin 26:7
Likewise, we find this ruling in the Tur and the Shulchan Arukh:
אסור לדון בפני דייני עכו"ם ובערכאות שלהם אפילו בדין שדנים כדיני ישראל
ואפילו נתרצו ב' בעלי דינים לדון בפניהם אסור וכל הבא לידון בפניהם הרי
זה רשע וכאלו חירף וגידף והרים יד בתורת מרע"ה
It is forbidden to reach judgment before non-Jewish judges, or in
non-Jewish courts (arka'ot) - even in a matter that they judge like
Jewish judges do. Even if both plaintiffs want to be judged by them,
it is forbidden, and all who come to be judged before them are
accounted wicked, and it is as though they insulted and cursed and
struck the Torah of Moses.
- Shulchan Arukh, Choshen Mishpat 26:1.
See the Rema (ibid.) for a situation in which you are allowed to use non-Jewish courts in order to place pressure upon one of the plaintiffs to desist, and see also the Rambam (ibid.) and the Tur/Shulchan Arukh (op.cit. 26:2) for situations in which non-Jewish courts can be used to place pressure upon Jewish plaintiffs who refuse to appear in a Jewish court. These texts speak of non-Jewish courts that are imposing financial, or possibly even corporal, punishments. They do not explicitly deal with capital crimes.
For that, one might look to the story recorded in the Tosefta (Terumot 7:20). There, the Tosefta mentions a situation in which non-Jews request a Jewish individual for slaughter, without whom all of the Jews in a particular group will be killed. Rabbi Yochanan holds that if they single out a specific person, that person should be handed over. Rabbi Shimon ben Laqish, however, holds that a specified individual should be handed over only if he has committed a capital crime.
The Jerusalem Talmud (Terumot 8:4) quotes part of this story, but then presents an example in which this transpired to Ulla bar Qushav. He was sought by the government for execution, and fled to Lod. Rabbi Yehoshua ben Levi, in order to appease the authorities who were threatening destruction on the city, handed Ulla over for execution. Thus far, this appears to represent the ruling of the Toseftan passage. Although it is unclear whether or not Ulla committed a capital crime, his being singled out specifically means that he should be handed over. Nonetheless, the passage continues by suggesting that Elijah ceased revealing himself to Rabbi Yehoshua ben Levi as a result (cf: the Pnei Moshe; the text is difficult here), who is then informed that the ruling from the Tosefta is only for the truly pious, while in practical terms one should avoid giving Jews over for execution.
This same passage occurs, with minor variations, in Bereishit Rabbah 94:9. One of those variations is that it makes clear in this version (as explained also by the Etz Yosef) that not only should one never give others over for execution, but one should also not deliver oneself. It therefore appears to be the conclusion of this particular narrative that Ulla was right to have sought refuge from the government, although it unfortunately still doesn't make clear whether he had incurred their punishment or not.
For more examples of situations in which Jews may or may not deal with secular authorities at all, see also Menachem Elon, Jewish Law: History, Sources, Principles (Vol. I; trans. B. Auerbach and M.J. Sykes; Jerusalem: JPS, 1994), 13-18. He brings various examples of edicts and responsa, permitting (under certain circumstances) the uses of non-Jewish courts to "decide on matters in which the general government had a special interest, such as real estate, the payment of promissory notes, government taxes, currency matters, and assault and battery" (op.cit. 17).
Not all Jewish authorities have accepted the use of such edicts, but amongst those who did are the Maharach (Rabbi Hayyim Eliezer ben Yitzhak, son of the Or Zarua), Nachalat Shiva (Rabbi Shmuel haLevi, 17th c. Poland), the Baal haNetivot (Rabbi Yaakov Lorberbaum, 18th-19th c. Poland), and the Rema. Those who disagree with such edicts include the Ramban, Rabbeinu Tam, the Rashbam, the Rashba, and (apparantly) the Maharam of Rothenburg. One example of a scholar who permits the use of non-Jewish judges, but only under certain circumstances (specifically, if both litigants name a particular non-Jewish judge whom they trust) is the Shakh (Choshen Mishpat 22:15). You would need to extrapolate from the situations that they describe in order to form a ruling that pertains to the specific case of which you are speaking.
Note: this answer was copied in large part from an answer that I gave to another question, and edited to fit this one.