Hi. I think the use of "could save" is correct because I think it could be used in a time context that is before the past (so to speak, if I say it right), but have a hard time thinking of the difference between the two (please look at the underlined part). Could it be the subordinate clause part of an adjectival clause (I think it is) "that it could have helped them" usually (or never???) stays in a past tense when the tense in the main clause of an adjectival clause(I think it is) is in a past perfect tense?
The people who had hoped that it could help/could have helped them were now convinced that nothing short of financial help could help them.