Heya,
Small wide primes for crop? Nothing will be that size, pancake. Wide, prime and small do not go hand in hand. One of the smaller primes that is kinda wide would be the EF 28mm F1.8. Otherwise, you're not going to be using a prime for what you're talking about. Almost all primes that are even remotely affordable for what you're asking to do, are not wide at all on APS-C. Even if you look at vintage glass, you're not finding wider than 28mm that are small.
So the only prime that will work here that is in your budget, is the Rokinon 14mm F2.8. It's quite wide, very sharp, fast, and will do what you ask, and be a prime with F2.8. It's not small. But no lens is going to be small that you're looking for. Pancakes are always in a certain focal length because of optical mechanics.
The 10-18 STM is in that price range, small but not pancake, and very wide, but slow.
The EF 28mm F1.8 is a relatively small prime, fast, but not really wide, more "normal" on APS-C.
Rokinon 14mm F2.8 does what you want completely. But it's not small.
It will all be a compromise no matter what since you asked for something that frankly doesn't exist. Not even for full frame, I'm afraid (pancake size wide prime).
I had the same issue, so I bought anEOS-M with a 22mm F2 pancake
. Small, wide enough (35mm FOV full frame equivalent, not ultrawide, but wide), fast enough for indoors, wide enough for big stuff outside. The whole thing costs less than any lens you're considering, and can be had for around $300 or less. Small system, great for walk around, still uses your other lenses.
Very best,