ok i've been on youtube for a while trying to figure it out and i think i've made some progress. here's what i got:
california dreaming - flute solo (16 bars):
Am Am Am Am
Am Am Am Am
Am Am Am Am
Am Am Am Am
C C E7 E7
Am Am F F
E7s4 E7s4 E7s4 E7s4
E7 E7 E7 E7
Am Am G G
F F G G
E7s4 E7s4 E7s4 E7s4
E7 E7 E7 E7
Am Am G G
F F G G
E7s4 E7s4 E7s4 E7s4
E7 E7 E7 E7
i'm not sure it's perfect but it sounds ok to me. the main doubt is whether bar 16 is E7 or E7s4.
@sophiehiker i can see why you play it the same as the main progression: it pretty much is: bars 1-4 of the solo are different, but bars 5-8 of the solo are the same as bars 5-8 of the main progression, and bars 9-12 AND 13-16 of the solo are the same as bars 9-12 of the main progression.

can't believe i couldn't hear this earlier. wow... first time i transcribed sth by ear and i see how justin's emphasis on it lays out now. it feels really good.