I'm sorry, I attached to wrong file for the Benedictus. Here is the organ accompaniment for the Benedictus antiphon:

What Andrew said, but also:

5. Is the second above the final a major or minor second?

Then
* if (5) the second is minor, it's 3 or 4; otherwise it's 1, 2, 5, 6, 7, or 8

I tried mode 6 with the Mag antiphon ... So 6 only satisfies one of the tests... but it still sounds good.

If you mean psalm tone VI with e (not c#) as final, reciting on g#, that's because modes I and VI have quite a lot in common, and even more so their psalm tones. But the antiphon is mode I. Obviously an adaptation of https://gregobase.selapa.net/chant.php?id=2486

Let's have a look at the first few antiphons in the 4-week psalter from the collection.
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EP1 Antiphon 1: No key signature so Do will be C, final on E = Mi, therefore mode 3 or 4 (Phrygian and Hypophrygian). While the melody stays above the final, it doesn't go very high, so mode 4 suits better; Reciting tone is on A = La, use ending E (ending on Mi) to end up back near the starting note of the antiphon ready for the repeat at the end.
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EP1 Antiphon 2: Four flats in the key signature, so Do will be A-flat. Final is on E-flat = So, therefore we're in mode 7 or 8. Tessitura isn't wide, and the melody briefly dips below the final, so I'd go with mode 8; Reciting tone is on A-flat = Do, use ending C (ending on Do) to end up near the starting note of the antiphon.
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EP1 Antiphon 3: Six flats (why would you do that???), so Do is G-flat. Final is D-flat = So, therefore we're in mode 7 or 8 again. Wider tessitura than the previous antiphon, I'd go with mode 7; Reciting tone will be A-flat = Re, use ending A (ending on La).
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Incidentally, for all three of these, the first reciting tone Gethsemani gives has matched the most appropriate Gregorian tone. Having looked ahead, that doesn't always hold, but it could be helpful. Let's continue.
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MP Antiphon 1: This has been allocated the same Gethsemani psalmtone as EP1 Antiphon 3. The same rationale holds.
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MP Antiphon 2: Four sharps, which makes E look like Do, however, with a final on C-sharp, that would make our final La. Generally, the finals are restricted to Re, Mi, Fa or So. This is resolved by setting the final to Re, and flattening Ti every time it occurs; we're in mode 1 or 2. Melody goes quite high above the final, so it's mode 1; reciting tone is G-sharp = La (and notice how the psalmtone has Ti flattened). Ending could be basically any of the options; I quite like ending A or A2 (ending on G-sharp = La), because returning at last to the antiphon, you get quite a nice consonance with the perfect fifth interval on 'midst'.
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MP Antiphon 3: Four flats, Do looks like A-flat; Final on C = Mi, so mode 3 or 4. Doesn't dip below final, reaches a sixth above, so let's say mode 3. The new Solesmes office books put the tonus 3 reciting tone on Ti (the ancient form), whereas older books put it on Do (aka tonus recentior). Let's go with the ancient form; Reciting tone is G = Ti, I suggest ending A2, but you could try others.
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That will do me for now! Hopefully my process sheds some light. There are some other ways you could think about modes to reach the same or similar conclusions. Finally, I'm certainly no expert, so others may disagree with good reason.

Another thing that may help: sometimes the music seems to be inspired by Latin antiphons with the same text, especially on major feast days. For example, at second vespers of Christmas, "You have been endowed", "With the Lord", "In the beginning" and "Christ the Lord is born today" are similar to "Tecum principium", "Apud Dominum", "In principio" and "Hodie Christus natus est" as found in Antiphonale Romanum (and other sources) respectively, and I would use the psalm tones given in Antiphonale Romanum.