She invited me to go with them, which I'd quite like to (do)

Solution 1:

When one searches in the Corpus of Contemporary American English for a sentence that ends with:

which (pronoun) (verb) to do (full stop)

vs

which (pronoun) (verb) to (full stop)

by far the more popular variation is the one with do added, especially when one only counts the examples where the relativised element is a VP, and the antecedent is an infinitival functioning as catenative complement.

The same is also true of the British National Corpus.

So, as far as acceptability is concerned, both seem ok. But, the construction with do would seem preferred, or at least, more common.