Correct: something with which we should not put up
Um, no. Correct is: "Something (which) we should not put up with." (One can say it with or without the "which", but in either case "put up with" is a fixed phrase and can't be broken apart.)

My strong advice is that "to put up with" is one of those phrasal verbs that SHOULD NEVER BE SPLIT.

On the other hand, the phrasal verb "to turn on" in the sense of включить is "splittable" -- you can "turn on the computer" or you can "turn the computer on" -- but you can only "put up with a problem". You can't "put a problem up with" or "put up a problem with", etc.

Another way to explain this is that the "with" in "to put up with" should NOT be considered a preposition after the two-word phrasal verb "to put up"; rather, it's the final part of a three-word phrasal verb that takes a direct object without a preposition. In fact, the three words in the phrasal verb are so inseparable that it would be rather logical to write the verb через дефисы: "How long must we put-up-with this idiot?" But, although it'd be logical, as it happens we don't hyphenate this verb.

P.S. Keep in mind that I speak U.S. English -- so it's possible that a speaker from the U.K. would disagree with me on whether it's correct/normal/natural to say "with which we won't put up."