I am presuming your bike is mostly stock. Your descriptions do lead me to believe it is a carburation issue. I suppose you have done the following:
1. verified good compression on both cylinders.
2. verified steady ignition timing (all plug wires).
3. verified both carb fuel bowl float levels.
4. verified good idle carb syncronization.
5. verified all hose connections to the carbs (nice and tight)
6. checked for vacuum and carb boot leaks with WD40 or water.
7. verified mixture screws set to initial 2.5 turns.
If all this has been checked, there may still be some obstruction to the pilot jet or pilot carb body passage in one of the carbs.
One last thought, you might try gently running the mixture screws in until they seat, and then run them back out 2.5 turns. Sometimes that will crush some junk out of the needle seat.
That's all I can think of. Maybe someone else will chime in soon.
Good hunting ...
'91 VN750 - Cosmic Splendor/Candy Wine
Done: Coastered, .020 ign., Iridiums, VM
Plexistar II, MCCT's, Relocated MOSFET
Will Do: Horns, Shave(maybe)