Mitchell
Another .02 cents worth.
There ate 3 places, and 4 valves that shut the propane off from the engine. A slow to start propane engine has a problem as it should start as fast as a gas engine.
The first place the propane of shut off from is the main fuel valve, either a vacuum operated VFF-30 or an electrical solenoid valve, that shuts the propane liquid down from the engine - the vaporizer.
The vaporizer has a primary (liquid - labyrinth- vapor) valve, a small valve operated by the little diaphragm inside the vaporizer. The secondary valve is as big as your thumb and that regulates the vapor - the mixer (the big hose to the mixer/carb)
The final place the fuel is shut off from is the cone (the thing that lifts as the engine speed increases) everything but an electric solenoid valve is operated by a diaphragm/valve, everything has to be leak free and not sticking.
Sometime they spray the vapor coming off of a refinery stack with anti-freeze (ethalglycal) to cool more propane out of the operation, sometime that anti-freeze gets delivered with the propane, but gums everything up, dish-soap and warm water is the best solvent to clean it.
In a nutshell, propane systems are a cake-walk to TS, you just don't wan't to tell anyone that you can use an electric screw-driver as there are just a lot of screws on everything, but taking things apart dose not mean you need a kit every-time,,,,, in fact it is more rare to need parts or kits just to clean things,,,,,, if you do find something wrong, you will loose the chip-truck till the part gets there, but will still be cheaper then shot-gunning parts at it.
Just being handy with tools, it is hard for me to think that it would take a whole 2 hours to go through everything on the system, twice.
My guess, clean the 2 valves in the vaporizer, under the huge diaphragm there is a tetertoter that activates the big secondary valve. Inside the secondary chamber there is a smaller housing and another lever of sorts that closes the liquid from the vaporise, that is your problem, or the first place I would go, a slow-to-start with back-firing is a lean system, and maybe sometime it just wont start?