If I were transcribing this, I would approach it just like you have--by tackling one bar at a time. Then I would follow this method.
Within each bar, I would use with the right hand melody/solo line to establish the nearest meter/beat. For example, I hear measures 1-2 as being in 6/4 time.
Then I would approximate simple rhythms for the melody/solo line, removing any interpretive stretching or squeezing of the rhythm.
Finally, I would transcribe the left hand and pair the left hand rhythms with the right hand rhythms.
There are ways to notate the "stretching" and "squeezing" of the tempo. For example, in measure 2, the left and right hands are off from each other rhythmically, and I think this is because the right hand is pulling back a bit, lagging behind the left hand. Nonetheless, I would still write those notes as simultaneous and would add the text "pull back" above the right hand part.
For example, this isn't perfect (and I haven't listened closely with any transcription software), but my first pass would be to notate it like this:
The third measure "squeezes" the time a little bit--it cuts out a beat and moves on to the fourth measure more quickly. For a case like that, I would simply notate measure 3 as being in 5/4 time. Alternatively, in those scenarios, you could notate it as 6/4 and put some text indicating to move quickly to measure 4.
The really fast notes in measures 4-5 might be a little quicker than sixteenth notes, but I would notate them as 16th notes since that's the nearest beat:
and perhaps I would add some simple text (maybe "play faster") indicating to speed up, though this probably isn't necessary. Too much of that type of text, though, can make the piece hard to read, so I would save it for the most extreme cases.
EDIT: There's another reason I mention the sixteenth notes--sometimes I have to find whatever touchstones I can to establish the beat/rhythms. I knew those notes in the right hand were faster than eighth notes, but it could've been hard to discern whether they are eighth note triplets or sixteenth notes. So I turned to the bass in the left hand and listened more closely to the second note (the D). This D sounded like it occurred on a downbeat, and this was crucial information: this base note D (this downbeat) occurred immediately after the fast run in the right hand. The only way for the fast run to conclude immediately before a downbeat (the D bass note) is if those fast notes in the right hand are sixteenth notes, not eighth note triplets. So that measure is an example where the bass note can serve as a touchstone for the beat. (It helps a lot that the bass/left hand is repeating a similar pattern throughout all of these measures.)
This is a really fascinating solo piano part to transcribe, and it provides a way to study several interesting notating/transcription techniques. Great choice!