I don't see how this is any different from using a proof published by another author in your thesis. Papers are published so that their results are cited. It would only be plagiarism if you didn't cite it appropriately. Your real question seems to be "can I include the proofs themselves in my thesis?" and it seems that the answer is yes. Think of what it would mean to re-write the proof - presumably you'd use different variables, but the logic and reasoning would follow almost exactly. If it didn't, then either it's a new proof (and therefore it isn't under copyright and may lead to another publication) or the proof would be incorrect.
I also think you are misunderstanding the point of the re-write. You state that "I'm not allowed a stapler thesis, so some amount of re-write is necessary". I disagree. Simply re-writing an entire paper and putting it into your thesis, even if you edit it for flow, doesn't really relieve you of the responsibility that you seem to have - that the results be either entirely novel or that there is a single, big result in the thesis rather than several smaller ones. If you have 5 loosely related papers and put it into a thesis, that's what we might call a "stapler thesis". If you have 5 papers that follow a logical progression from part 1 to part 5, then whether you just staple the papers together and call it a thesis or you write the thesis in 5 chapters, ensuring that the ideas/writing style flow properly, that's a different thing altogether. However, if there is a mathematical proof in one of those papers, re-writing it won't change the fact that it's already in that paper and therefore is not a novel result.
If your thesis depends so strongly on those proofs that the thesis essentially is the set of proofs, then I'd argue that unless you did that work as a PhD student you'll have a harder time convincing people that your work is worthy of a PhD - it wouldn't be novel research. If we could do that, then a well-published author could just enroll in a PhD program ever year and complete it by just submitting a thesis made up of the results from previous papers. On the other hand, if these proofs are useful, but an uninterested reader could just accept the citation and an interested reader could go and find the paper, then I think you have nothing to worry about by including them.