"Don't let's" is theoretically correct in English and I can think of it being in use until the 1940s or 1950s but it is definitely dated now. It is not used except in English regional dialects (some Yorkshire or Lancashire branches of the language would use it, but then again they use 'thou' as well) and in comedy or sardonic speech. In fact since 'let's' has come to convey quite a positive, pro-active emotion nowadays "don't let's" almost sounds like an apposition.
Use "let's not" in all common speech! :) Let's not get confused ;)