I quite often hear people saying e.g. "I don't know as well" or "He is not joining as well".
Apart from being "more neutral", is it grammatically correct to combine "not" with "as well" (or also, too)?
I have some difficulty believing you hear such constructions "often," unless you work with non-native speakers, as they are, indeed, ungrammatical, and not the sort of grammatical gaffe most native speakers typically make. (As DH suggests, using the wrong word here simply "sounds wrong.")
At any rate, it is a rule: a negative statement requires the use of either+ . +Too+ , +also+ and +as well are reserved for positive statements.
The exception is when positive and negative statements are combined in a single sentence:
He drinks too much, but at least he doesn't smoke as well.
or
You can have an apple, but you can't have an orange too.
(examples borrowed from Swan's Practical English Usage )