So UDT seems to eliminate time-inconsistency, but at the cost of increasing the number of possible outcomes, essentially turning games with sequential moves into games with simultaneous moves, with the attendant increase in the number of Nash equilibria. We're trying to work out what to do about this.

Er, turning games with sequential moves into games with simultaneous moves is standard in game theory, and "never cheat, always punish cheating" and "always cheat, never punish" are what are considered the Nash equilibria of that game in standard parlance. [ETA: Well, "never cheat, punish x% of the time" will also be a NE for large enough x.] It is subgame perfect equilibrium that rules out "never cheat, always punish cheating" (the set of all SPE of a sequential game is a subset of the set of all NE of that game).