Is a rillette typically cooked in fat?
Rillette is pork cooked in fat (not really confit, but close to).

Most of what I've seen on the internets is that you can freeze the rillette for longer storage; if not, keep at most for 30 days in the fridge.

You can pour a little bit of the fat on top of the rillette in the container, mostly for extra flavour, not for long term conservation.

I don't know enough about pink salt (Himalayan salt or the Pink curing salt ?) to help you there.