Q:how would that affect your choice of CAN or COULD?
A:I had the question because I thought consistency in English grammar is important, and this might forbid CAN, which sounds too 'present tense' to be used with WERE, which indicates a hypothetical situation, which traditionally requires the past.
If there were one virtue I can/could claim, it would be punctuality.
This isn't an irrealis hypothetical, WB, in the same sense as "If I were you, WB, ... .
The choice of 'were' here isn't absolute. Are you aware that 'was' can be used for counterfactuals/irrealis?
This sentence is the same as saying, "I don't want to seem like I'm bragging so I'll put it in doubtful terms that I think my best virtue is punctuality".
We use 'If + were' quite often to make soft suggestions, to give advice in a not too forward fashion --> "If you were to do it this way, it might be better".
English speakers never seek consistency over meaning. That sounds like advice coming from those that believe English has Tense Concord/Sequence of Tenses. It doesn't. We have to be able to say all things, to effect many nuances. If you believe that modal verbs have tense, you are, as an ESL, missing out on a lot of meaning.
False rules like this "modals have tense" do not affect English speakers because our internal grammars know what the real rules are.
CAN, as the historical present tense [no longer true] has kept it meaning as more REAL, more POINTED, more DIRECT than historical past tense [no longer true] COULD, which continues to be more INDIRECT, more ETHEREAL, more DEFERENTIAL, ... .