Is "It won't let me not." grammatically correct?

What you're saying is perfectly grammatically correct, and in my opinion it's completely natural. All you've committed is ellipsis of the phrase fasten my seatbelt:

I always fasten my seat belt because my car won't let me not fasten my seatbelt.

This is different from the case with allow, because the verb to let takes as its complement a bare verb, without the infinitive marker to.

I'll let you leave. (wrong: *I'll let you to leave.)

The nurse won't let you not take your medicine. (wrong: *The nurse won't let you not to take your medicine.)

So when using ellipsis with the verb let, it's correct to leave off the infinitive marker to, and to place the negative not at the end if required.