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.
If you do that, why wouldn't you use a FPGA or just a fast CPU? Microcontrollers and CPUs are blending in performance, and are cheap enough to plop on a board and call it done for many applications.
Sure, but if really want to avoid trusting trust (and you're of the mind to build your own hardware), FPGAs and µcs offer a lot of room for snooping.
Given the GPs suggested use, it seemed trusting trust was not on the table.
Certainly even a tiny FPGA can fit pretty naïve versions of common crypto primitives, as can any modern micro-controller. Assuming you only need to do a handful of ops for whatever you're looking to assert/verify, that is by far simpler than building a gate-level representation :)
I was thinking about a chip with only sram for secret storage that could be bundled into a ID-1 sized card with some small energy storage for the sram (there are affordable .5mm LiPo Cells that fit inside such a card), and then use the card to fit some display capable of giving some little data out, as well as a touch matrix,possibly by just using a style similar to carbon-contacts on cheap rubber membrane keyboards, but gold plated like the smartcard interface. But it seems like you can't afford to store one decompressed ed25519 or dare rsa, so the idea is moot by virtue of requiring sub-100nm technology to fit at least some sram.
In the usecase a lack of accessibility of sram content through probing is very important. The benefit of this over some μC or fpga is that you can account for every spec on a scanning tunnel electron microscope. And the high resolution xray you made before while the chip was still in it's package. Which you can compare with all the chips you will use and ship. It is sadly easy to backdoor with just a single hidden gate.
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.