I can’t seem to find any rings with Mifare 4k. Are there none, and if so why not?
The demand for a 4K ring is smaller and it’s harder to source magic 4k chips especially in a form factor that would fit in the ring.
Have you had a look to see whether or not you actually need all 4K of storage? In some cases the data on a 4k if not exceeding block 63 can be truncated down to fit on a 1K ~ also depends on the length of your UID
Have you thought about making one?