The correct answers are We needn't have brought the laser cutter. and We didn't need to bring the laser cutter.

As a marginal modal, need can act like a true modal auxiliary or a regular lexical verb.

"Needn't have brought" uses "need" as a modal (no do-support, bare infinitive). In the past, needn't have + past participle specifically means the action was performed, but it was unnecessary.

"Didn't need to bring" uses "need" as a lexical verb (takes do-support, uses a to-infinitive). It simply states there was no obligation, regardless of whether they actually brought it or not (though in this story, we know they did!).

"Needn't to have" and "need not to bring" incorrectly mix modal syntax (no do-support) with lexical syntax (the to-infinitive).