Commercial Bakery to Provide Dough
The best solution would probably be to get the buns par-baked, or to get frozen dough. 
The problem with buns is that they're small and they rise very quickly. At a commercial bakery where I worked, most of our buns would be ready to bake within 2 hours tops, even if they spent their entire second rise refrigerated.
Frozen dough can be easier to work with, if it's frozen correctly. If it's mixed cold and frozen as soon as possible after mixing, it can work pretty well. If it's frozen too late, the yeast will be too active and cause a boozy flavor when they're baked, and the extra yeast die-off can make the dough go extremely slack when it thaws. Getting frozen dough leaves you with the problem of keeping enough dough thawed (and then being sure you can get the thawed dough molded, proofed, and baked in time, which you'd have anyway, frozen or not).