Major edit after OP's clarification
I'm pulling nomenclature from a paper written by Myles Skinner, a microtonal community wiki, and Wikipedia. I'll refer to quarter tone intervals as decimals between the semitone intervals.
3.5 is pretty universally called a neutral third. That's from all three of the sources and personal experience. It's a good representation of the just interval 11/9.
6.5 on the other hand, is called a narrow fifth on the microtonal wiki and a minor fifth in the paper and on Wikipedia. Either way, the nomenclature works. Narrow inverts to wide and major inverts to minor. For the sake of using familiar terminology and at the risk of further overloading the words, I'll go with with major and minor. It's a good representation of the 11th subharmonic, 16/11. It's also called a "wolf" fifth, which is a term borrowed from mean temperaments and pythagorean temperaments where the lack of a closed circle of fifths made one "fifth" particularly out of tune.
9.5 isn't even referred to in Skinner's piece, but the microtonal wiki says it can be called an ultra sixth or an infra seventh. They're enharmonic and both represent 26/15 very closely. Wikipedia calls those intervals subminor or supermajor and says they approximate 12/7. For the structure of the chord, we'll go with the seventh and for sticking with major and minor, call it a subminor seventh.
So, you could call this a G+nsub7m5. What a chord and its name is fully Wikipedia-compliant! Regardless of what you call it, here's what it sounds like:
- The flamenco scale with the G+ in it
- The chord arpeggiated across two octaves and then played across one
- The chord used as a V in a V-i progression
- The chord used as a V in a V-I progression
- A different voicing of the chord in a vii°-i type setting
In my opinion, it sounds much better as a diminished chord. If I had more time, I'd invert it, try to modulate, maybe compose with it a little. A chord in between a diminished seventh and a dominant seventh chord is definitely fascinating, but this is all time permits for this answer. Hope it helps.