british english - 'to'-infinitive without the verb

I seem to recall reading somewhere that using a to-infinitive with the actual verb omitted (because it's clear from context) — as in

He asked me to go, but I don't want to. (1)

— is fine in American but not in British English. Brits, or so the story went, append do:

He asked me to go, but I don't want to do. (2)

I know that the above is true about American English, my native dialect: we can use (1). My question concerns British English.

Googling finds that the above (that (1) is wrong in British English) is not correct in such generality. For example, "can't be arsed to if" has fifteen-odd results, while "can't be arsed to do if" has but one, and it's not in the form of (2).

So...

Did I imagine the rule I stated above? Or is it restricted to particular sentences (or verbs or something)? Or is it correct as stated but outdated? Or what?

Answer

I find that this different in use of "do" between British and US English is more common with auxiliary verbs, not infinitives: "I didn't take the garbage out, but I (should have/should have done)." Then again, I'm not British. I may have interpreted this incorrectly.