OK, it turned out to be something fairly obvious -- we were calling this as a Read (GET_ENTITY) when only Query (GET_ENTITYSET) was set up (this was not obvious, because none of this was set up in SEGW, it's all custom methods). Somehow I got the impression from the web side that the Travel app was calling a Read but now I'm pretty sure they are only doing the Query. Apparently within these non-SEGW implementations, sometimes even the generic READ method to find the correct entity READ method doesn't get called, let alone the actual READ entity method in the DPC_EXT class.
So, at least I learned that the "resource not found" error can result from SAP looking for methods that are not implemented (as opposed to returning the "not implemented" error from within the method itself).
Leaving this here in case it helps someone else!