Is there documentation to prove that the Aramaic word "דילמא" is a transliteration or derivation of the Greek word "dilemma"?
If so, this fact is informative to G'mara-learning in understanding the precise connotations of terms.
From Jastrow, page 299:
דִּילְמָא, דִּלְמָא I (= דִּי לְמָא, v. Ezra VII, 23; = h. שֶׁמָּא = שַׁלָּמָה) for why, whence 1) lest, perhaps. Targ. Deut. VII, 22; a. v. fr.—Ab. Zar. 35ᵃ ד׳ איכא וכ׳ lest there may be one who &c. Ber. 29ᵇ מסתפינא ד׳ וכ׳ I am afraid, lest I may become confused; a. fr. —2) (without the meaning of apprehension) perhaps, it may be. R. Hash. 3ᵃ; a. e. כִּי has four meanings: אי ד׳ אלא הּא if, perhaps (lest), but, because. Ber. 2ᵇ top ד׳ ביאת וכ׳ is it not possible that the word uba indicates the arrival of his sun (the morning of the eighth day)? Ib. או ד׳ or may it not be; a. v. fr.—[Pesik. Shek. p. 13ᵃ דד׳, corr. דילמא, as Tanḥ. Ki Thissa 5.]
This seems like a better, and more straightforward, etymology. There is a clear basis in Ezra 7:23: כג כָּל-דִּי, מִן-טַעַם אֱלָהּ שְׁמַיָּא, יִתְעֲבֵד אַדְרַזְדָּא, לְבֵית אֱלָהּ שְׁמַיָּא: דִּי-לְמָה לֶהֱוֵא קְצַף, עַל-מַלְכוּת מַלְכָּא וּבְנוֹהִי. 23 Whatsoever is commanded by the God of heaven, let it be done exactly for the house of the God of heaven; for why should there be wrath against the realm of the king and his sons?
where it is clear that di+le+mah is 'that + for + what'. And also the parallel to the Hebrew cognate shema = she + la + ma. Of course, words shift slightly in their meaning with use.
I must compliment you on your good eye in spotting the Greek word with similar meaning. It might be useful as a mnemonic, as often dilma can be used to speak about two possibilities.
But ultimately, I am fairly certain that it is a false cognate:
False cognates are pairs of words in the same or different languages that are similar in form and meaning but have different roots. That is, they appear to be, or are sometimes considered, cognates, when in fact they are not.