Not for an ASIC without spending a LOT on tooling, and really $10k is awfully optimistic even if you had all of that tooling (I probably should've just said tens of thousands).
For <100k, yes, you can absolutely do a small run in that range.
Honestly, you might be better off just buying functional ICs (multi-gate chips, flip flops, shift registers, muxes, etc.) and making a PCB, though. Most crypto stuff is small enough that you can do a slow/iterative solution in fairly small gate counts plus a little SRAM.