Whoa, wars have been fought over what I am about to say.
First, what is a mode. It's a mood of a major scale. Mixolydian and Lydian are moods of a major scale. OK what major scale are they a mood of? Any one you like.
Scale degrees:
1, 2, 3,.. 4, 5, 6,. 7 is the major scale.
1, 2, 3, #4, 5, 6,. 7 is the Lydian mood of the major scale.
1, 2, 3,...4, 5, 6,.b7 is the Mixolydian mood of the major scale.
The Lydian dominant mode will have both a #4 and a b7. But, it is still just a mood of a major scale.
Which major scale. Any one you want. The one you chose is what decides the key signature. Key signatures show what major and/or minor scale/key the song is to be played in. Not what mode is to be used to play the song. That is dictated by the notes used. Yes you will go out of scale to get the mood you are looking for - you sharp the 4th for Lydian and you flat the 7th for Mixolydian. For Lydian dominant you will use both the #4 and the b7.
If you want to use Dorian you just moved from the major scale to the natural minor scale. How would this affect the key signature? I don't need an answer, you hash it out....
Natural minor scale 1, 2, b3, 4, 5, b6, b7.
Dorian takes the natural minor scale and sharps the b6.
Phrygian takes the natural minor scale and flats the 2.
Locrian takes the natural minor scale and flats 2 and the 5
If you are still using:
Ionian = C, D, E, F, G, A, B
Dorian = ...D, E, F, G, A, B C
Phrygian =.....E, F, G, A, B, C, D
That's relative modes; easy to teach thus what most of us were taught. Easy to teach, hard to use. Parallel modes are much easier to use in a song. Parallel modes are; want Phrygian, use the natural minor scale and flat the 2, if you want Dorian use the natural minor scale and sharp the b6 back to a natural 6.
Good luck and have fun.