Learn English – Can you use “get” in a past tense sentence

In your first sentence, you switch from past to present tense where the second clause is a temporal qualifier, which is not allowed.

"...he comes over quite often, he and I got on quite well when we see each other so not much issue there."

A more extreme example of this not being right might be, 'I was tired when I am hungry'.

Your second sentence is correct, because the tense matches.

To answer your title, which is broader, you can combine tenses in a sentence if they are in separate clauses but remember to match tense when you use 'when'. An allowable example without 'when':

I have to get my hair cut because I was told that it is too long by my teacher.

There is an incorrect assumption in the question.

Someone seems to have promulgated a rule saying that you can't use present tense and past tense in the same sentence. This is not true -- indeed, it's silly, and shows that wherever that rule came from is not to be trusted on English grammar.

In this particular case, conjunction reduction deletes the second which from the relative clause (which) makes me feel .., but that has nothing to do with the tense of the verb in the relative clause.

Executive summary: There is no reason not to use those tenses, if that's what you mean.
And you can certainly use all kinds of tenses and constructions together.
Provided you follow the rules. The real rules, that is.