You haven't quoted the full sentence! It is:
It was a matter of chance that I should have rented a
house in one of the strangest communities in North America.
The should-less version would more naturally be "that I had rented" rather than "that I rented". I agree there would no real difference in meaning.
I also agree that "have" here is a bare infinitive: modals like "should" are always followed by bare infinitives - we say "she should have", not "she should has"!
The modal "should" here serves a quasi-subjunctive function here emphasising the chance nature of the course of events.
"Should" does not always express regret or hope. Sometimes it performs quasi-subjunctive functions, e.g. "The law required that he should do it", which means the same thing as "The law required that he do it".
Also, in some varieties of English (such as southern British English), "I should" can simply mean "I would". ("I shouldn't be surprised" = "I wouldn't be surprised".)
Definitions 3, 3.1 and 4 at Lexico are particularly relevant:
3 formal (expressing the conditional mood) referring to a possible event or situation.
‘if you should change your mind, I'll be at the hotel’
3.1 (in the first person) indicating the consequence of an imagined event.
‘if I were to obey my first impulse, I should spend my days writing letters’
4 Used in a clause with ‘that’ after a main clause describing feelings.
‘it is astonishing that we should find violence here’
There are longer lists of examples there.
You just have to use the context to interpret the meaning.
Here, "It was a matter of chance that..." is an important clue to what is meant. It would be fairly rare to say "It was a matter of chance that I ought to have..." or "It was a matter of chance that I was obliged to...", so a "should" of obligation would also be unusual here - and you can couple that knowledge with the fact that he has in fact bought the house, as you said.