I'm not really looking for an "objective" answer, but a list of pros and cons for the 2 options I listed (and maybe other options like notating the key as 3 flats or making a weird custom key). It would help to know what you think is best. It would also be helpful to know what the answer for this type of question be for other scales.
Due to how often both the Phrygian Dominant scales and the double harmonic major scales are used when soloing on the dominant of minor keys - often even regardless of genre, and this includes classical music! - I actually generally use minor key signatures instead of major key signatures when notating music that uses any of those scales. Also a golden rule: I generally do not switch key signatures with rapid scale changes as long as the tonic does not change. (Examples of rapid scale changes to and away from Phrygian Dominant scales include 0:51-1:46 of Dream Theater's "Ytse Jam" (I'd use a 1-flat key signature throughout that excerpt, excerpt swaps between A Phrygian Dominant and E Phrygian Dominant, the excerpt "soloing" on the dominant (of D minor) should be apparent) and Morpho Knight's cutscene and boss themes from Kirby and the Forgotten Land (I used a 3-flat key signature throughout this piece, other transcribers used 4-flat key signatures partially due to a problematic starting trill on C and D flat - piece uses C Phrygian, C minor, and C Phrygian Dominant, possibly among others)).
But for pros and cons, I may as well write this (assume all mentions of tonics and subdominants are of the parent Phrygian Dominant or double harmonic major scale):
Minor Key Signature of Tonic
- Makes the tonic clear
- More forgiving with rapid scale changes later (e.g. use of ^2 instead of ^♭2, use of ^♭3 instead of ^3)
- May involve more key signature changes than necessary, especially for dominant preparations of sonata-allegros and other soloing on the dominant (e.g. soloing on the dominant of D minor would involve using a no-accidental key signature here and a 1-flat key signature later)
Minor Key Signature of Subdominant
- Aligns with customs for notating modal music, especially music that uses the regular Phrygian scale (at least customs that involve notating as few accidentals in measures as possible and accurately reflecting the contents of the base scale)
- Is the most appropriate for dominant preparations of sonata-allegros and other soloing on the dominant due to reflecting the actual home key of those passages - see these 2 examples from classical music: Introduction of the Danse Bacchanale from Samson et Dalila by Camille Saint-Saëns, piano transcription by the composer - uses the A double harmonic major scale Dominant preparation of the 3rd movement of Ludwig van Beethoven's Piano Sonata No. 14 in C Sharp Minor, Op. 27, No. 2 "Moonlight" - melody of the 2nd to 2nd last measures uses the G sharp Phrygian Dominant scale - so do all of the 9th to 16th measures
- May obscure the tonic (this is important for music that primarily uses Phrygian Dominant and/or double harmonic major scales and never seems to "leave soloing on the dominant", such as "Wicked Thing" from Bravely Default)
Major Key Signature of Tonic
- Makes the use of ^(♯)3 clear (instead of ^♭3)
- Also makes the tonic clear
- Often ends up requiring more accidentals on notes in measures than other options
- Involves a later substantial key signature change if used for dominant preparations and soloing on the dominant (e.g. soloing on the dominant of D minor would involve using a 3-sharp key signature here and a 1-flat key signature later)
I would notate C double-harmonic major as though it were C major (no sharps/flats) and then use accidentals to indicate the modifications for the actual scale. (To clarify: for D double-harmonic major, I would use the D major key signature, and for X double-harmonic major, I would use the X major key signature.)
For C phrygian dominant I would use the C minor key signature (three flats), because phrygian is often thought of as derivative of minor, then use accidentals for the modified note.
The reasoning behind these choices is that modes are often conceived as modifications of the more familiar major/minor. And in particular, I'm thinking of scales like melodic and, especially, harmonic minor. Those aren't indicated directly by the key signature, but an experienced musician immediately recognizes their use by the presence of specific accidentals on specific scale degrees.
Since the double harmonic major is in fact still a major scale, it needs the key signature of the original key - for C, no sharps or flats. Then use accidentals to lower the 2nd and 6th notes. You can't really have a key signature that reflects that. At least one that would be recognisable to most, perhaps excluding Bartok!
The Phrygian dominant. It is based on the 5th mode of the harmonic minor, which is, some would say, an altered scale in its own right! Again lowered 2nd and 6th, but called dominant due to being based on the 5th note of the parent scale, (hence dominant, with M3) a key signature wouldn't make a lot of sense to most musos, so accidentals are order of the day.
Bit like there won't be a key signature for the chromatic scale.
Since you're not looking for the objective answer:
Two possible ways of thinking about the Phrygian dominant are equivalent in notation:
- Think of it as the fifth mode of a harmonic minor and use the same signature you'd use for that harmonic minor. Put an accidental to raise the third when it appears.
- Think of it as a Phrygian mode with a raised third.
In either case, C Phrygian dominant would use the key signature for A♭ major or F minor (4 ♭s), but with E♮ when it appears. I've also tried channeling Bartók and using the non-standard signature with B♭, A♭, and D♭, but that can be easy to misread since we see 3 ♭s and expect they are B♭, E♭, and A♭.
If I'm writing in double harmonic, I might notate with the signature of the major and add flats on the 2 and 6, (C major with D♭ and A♭ accidentals) or as the Phrygian with raised 3 and 7 (C Phrygian/4 flats with E♮ and B♮ accidentals). It depends on the context, and this scale is more often used with chords outside the scale.