There's no town on Caye Caulker, just a village, and everything on the island is right there.
Most places will take you for one night -- just depends on whether they have space.
Lots of nice, quiet places. Among them:
Treetops -- spic 'n span rooms for around Us$40
Iguana Reef Inn -- the Ritz of Caulker, nice A/C suites with queen beds, around US$100
Shirley's and the Anchorage -- both of these are at the south end of the island and though they are near the airstrip are quiet.
Even the places in the thick of the action, e.g. Trends Beachfront and Seaside Cabanas, both at the main public pier, really aren't noisy.
--Lan Sluder www.belizefirst.com