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For instance, if rejected from GDC LA, then apply to GDC NYC? Is there any potential downside to doing this when the offices have different recruiting managers and hiring partners (other than I'll obviously get auto-dinged if they keep any records at all)?

They'll say no more vociferously? I don't see how there could be a downside.

I also don't think your conclusion about being auto-dinged is necessarily correct. Maybe and auto-ding if you left an affirmatively negative impression (either through your credentials or interviewing), but I don't think it necessarily is. You could have been dinged at one office for a whole host of reasons (bad fit, not up to GPA standard, etc.) that may not be a concern with the other office (different office culture, lower hiring reqs., etc.). Not saying that is the case here but given the lack of downside, why not apply?

Why waste the time thinking about it and reading this...just do it and see what happens.

I've been dinged at one office and gotten a CB at another for several firms. You have nothing to lose by applying.

(Assuming cover letters were sent) know that they will have your cover letter from the other office, which might profess your undying love for another city.