Yeah, do that.

I had one from a pretty small piece of glass (less than a couple mm's wide). Booted and replaced tube, and the next day after riding home on it, the spot was bulging and the puncture was stretched. Yeah, it was Zafirro and you have a Gatorskin, but the tire is now structurally weaker.

But sure, test the waters and find yourself stranded after the tire eventually (and it will) fails on you.