Great question — answering as the UphillTech team:
My inclination would be to carefully remove the existing escutcheon around the sprinkler head if possible, or make an oversize hole in the replacement wallboard and cover the gap with a split retrofit escutcheon, such as these at Quick Response Fire Supply or these at Grass Seed Professionals (listed only as examples) . Be sure that any replacement complies with local fire codes and is compatible with the sprinkler system. Your idea to fill the gap with two pieces of wallboard, or a board with a slot cut out and a small strip of wallboard, would also work well. A large sheet of paper could be taped over the ceiling, and used to make a template for cutting. Since there seems to be no access...
Full answer by DrMoishe Pippik: https://diy.stackexchange.com/a/333603 (CC BY-SA 4.0).
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