Did the aircraft designation start with a B and end with a number?
Nope. PB4Y, P2V, DC-3 (an addition), C82, C97, etc.
and were there large doors fitted to the underside?
For the APU, no. Just a small exhuast. For mission purposes of dropping things, yes.
No need for bleed air from the APU?
- how would you start the engines?
- how would one dry crank an engine in case of tail-pipe fire?
An lot of aircraft don't have APU's to begin with, and don't need bleed to start, using electric starter-generators. Or separate starters and generators. So far as a tailpipe fire...if no bleed is required to start, lack of it from an APU has no bearing on continuing to crank during the start process.
I believe that the heat generated by a piston engine would require extra drag for cooling.
Cooling drag is irrelevant for ground power, and many aircraft do not or cannot use their APU in flight. We could use our piston APU's in flight, with no additional drag penalties.
The other factor is creating a piston powered apu that could maintain the 400 hz required for the generator and reliable power to do so.
Applicable for those aircraft which use 400 Hz...which not all do, and for those which do, stable inverter output can be regulated as needed.
There was a gent, in the '70s who was running around in a B25 with a Briggs and Stratton 4hp 2 stroke engine to supply electical power.
While some operators do use commercial generation, more than likely he was using the standard four stroke APU, which was generally a two cylinder carbureted piston motor with a direct drive generator.