Have you considered one of the Odroid boards from Hardkernel?
They also hav AMlogic and Samsung Exynos and others, but some are above your budget.
The Odroid boards are mighty if you want processing power. The XU4/XU3 have the same SoC as the international version of the Galaxy S5 are only about 2.5 times slower than an Ivy Bridge i7-3770. The XU4 is only 32 bit, but is 1.7x faster than the 64 bit C2. The C2 is 1.55x faster (on my tests) than a Pi 3 which is also quad-A53s. And Hardkernel supply a 64 bit Linux for the C2, whereas the official OSes for the Pi 3 only use its 32-bit compatibility mode.
So the overall difference between a Pi 3 and i7-3770 is just over a factor of 6. Oh, and I make the Pi 3 1.6x faster than the Pi 2 on my usual test, for almost exactly 10x slower than the old i7. New top end i7s are 2x faster than the 3770.
If you're in for an adventure, just maybe it's worth to look into the new Risc-V stuff. It seems that samples are becoming available and it seems that it's mostly sold out because everybody is jumping on it.
I think almost everyone is still just making Multi-Project Wafers (aka shuttle runs) in which you only have to pay the costs of a small fraction of the mask, but you only get 100 chips from each mask so the setup costs are relatively small ($30k maybe) but the individual chips are very expensive.
I expect multiple companies will be moving to volume production on various devices this year.