@jdjazz has given a better answer than I could, but I thought I'd add a few thoughts I had with this great question.
I was first occupied with getting away from the chord naming's from the auto program and instead focused on expressing each degree of the scale as a quartal 1, 4, 7, 3 (13, 9, 5) chord, with alterations. Which would end up giving all the modes of the scale in the extended quartal chord form. I deleted the text file I had for that but let me know if you'd like to see it as it would be easy to do again!
As the scale has both elements of high symmetry and also large gaps that impede symmetry it ends up with a curious combination of quartal chords and I loved the sound (on guitar, for me, but bass is my main instrument so I'm not so quick at seeing equivalent chords I may be oultlining on guitar).
For useage, I was thinking along the same lines as jdjazz. I was recently approached by a student who wanted to have an easy way of knowing what chord every common triad over every common bass note created, and we surmised it was too large a pool to really find a coherant method with which to work, but I encouraged him to just play a bass note (easy if you choose to make that an open string on guitar, initially), and then play a given triad off of different roots on top, until something stood out as interesting sounding. Then we worked out what the intervals for said chord were, worked out where they may be used as a substitute in diatonic harmony and used it to increase voicing vocabulary in a quite inventive and free way. This struck me as similar to how you may use double harmonic major chords, play, pause, evaluate and incorporate. In this case you are just looking at how that chord may relate to something in a more standard diatonic progression and use it as new sound you may be able to use. This may be more useful to a guitar player, where sometimes new voicings are harder to 'see', because of the fretboard limiting note choice, whilst also providing duplicates of certain notes that allow voicings of certain chords to appear in sometimes unfamiliar positions.
I wouldn't worry too much about taking common ii V I progressions etc. and applying them to double harmonic major, those progressions are important in diatonic use-age because of the diatonic major scale's construction, reconstruct them with a very different scale and they are unlikely to make as much sense as they do in major. But it's worth trying them to see what you get.
If composing, rather than finding ways to apply over existing diatonic harmony, you could also make use of some of the great sounds the scale gives, treating it as a parent key similar to how we do major or minor, but in that case the relationships within are very very different to diatonic and will lead you in new ways. So I would say work with finding the interesting relationships in the scale itself, such as the base quartal chord of the 1st and 6th degrees being the same, and using that as a jumping off point for composition in some section of a piece, with the option of then blurring it back into some kind of diatonic progression via how those chords could be interpreted against a major scale if desired..