Writing in third person past tense. I've already mentioned that a store has changed owners several times, when I then go to this sentence: "He leaned against the counter, smoking and listening to the radio with Bob, the current owner of the shop." Except the word "current" seems to imply present tense, when really I mean at the moment this story is being told, in the past. But I feel like if I were to say something like "...with Bob, the owner of the shop at the time," it makes it sound like Bob no longer has ownership, which is not necessarily the case. So can you use "current" to mean at the moment, but in the past?
I'd just omit it--it doesn't seem necessary to specify. "... listening to the radio with Bob, the owner of the shop."
I would say 'yes, current definitely' works and it doesn't give a feeling of mixing tenses. 'Current' is, after all, an adjective, and adjectives don't imply a certain tense by themselves (like verbs can do with different conjugations).
I'm with @elynne. I would just drop it altogether. Simply saying "the owner of the shop" already implies that he's the owner contemporary to the telling of the tale. Specifying it overmuch isn't needed, and as you have already discovered, is proving unwieldy.
There can be no doubt that it works both grammatically and syntactically. I was aiming my answer more toward your personal concern that it might lead to a temporal misdirection, hence my agreement that it can simply be done without.
