Is Caravan meant to have the drum solo, in-universe?

In-universe, it's improv, spur of the moment. No-one else knows he's going to do it and you see initially Fletcher's annoyance as the lights dim.

The rest of the solo builds Andrew's independence from Fletcher's overpowering rule that he's had over the duration of the plot.
His "I'll cue you" is a sure sign he's taken control of his own destiny as far as this band and his part in it is concerned.

The drop to white noise is a hark back to the fast swing^[1] he's been trying to perfect for half the movie - which Fletcher has always beaten him down on.

Once he's proven himself, he wins over Fletcher; who of course as band leader... and being Fletcher - insists he wants to regain control over. He demonstrates this with his conducting the rallentando [slow down] then accelerando [speed up] and finally the band comes back in for the triumphant last chord. We'll just suspend disbelief over how they knew exactly what to play; it's a movie, plot says that's what happens.

They've achieved a mutual respect, perhaps, by the end.

The whole structure of the solo out of universe, however, is clearly designed to be emotive in its content. Though it's a drum solo, it's really a character interplay almost without dialog. If it wasn't, a non-musical audience would be bored to tears after a couple of minutes of it.
As such, in my opinion, it succeeds in making a movie about music one of the few to be convincing even to a musician.
That it doesn't follow any existing solo is not surprising. Solos tend to be entirely improv, even if the overall feel or structure was pre-determined by the original player. They are rarely as plot-driven as this one though. It had to go through several emotional phases in the eye of the audience, not just entertain a concert-hall full of actual jazz fans.

^[1]
I can't actually remember how they described it in the movie, whether it was fast swing, be-bop, or whatever - but it was Andrew's "Feel the Force, Luke" moment, the struggle to overcome, to learn how to 'use the Force', to win the approval of his mentor.