"Let's not go there" or "'let's don't go there"

"Let's don't" is certainly informal, and the grammar is quirky, but I think it's dogmatically prescriptive to say it's incorrect. It's not a truly rare spoken usage either, which suggests it's doing some kind of work that "let's not" isn't doing.

One thought is that it can imply a certain location or context in an ongoing dialog that "let's not" doesn't capture. For example:

A: Shall we go to the park today?

B: Oh, let's do!

C: Let's don't.

Here, C can be read as explicitly regarding and challenging B. As a speech act, "let's don't" seems to have a parameter that the preferred usage lacks. Moreover, this dialog is perfectly intelligible and nothing hits the ear wrong as far as I can tell.

Another thought is that the "let's don't" formulation deliberately conflates the normal structure of a hierarchical injunction with a more egalitarian proposition, either to borrow authority from the injuctive structure, or to minimize the rank of the speaker for whatever reason, e.g. the boss says "Let's don't miss our deadlines, team."

In general, "let's" and "don't" are in tension because one admits the possibility of refusal and the other doesn't. "Let's don't" might either be a "Let's..." statement with heightened emphasis, or a "Don't..." statement with a kind of honorific egalitarianism.