I have a Toyota 7fgcu45 doing basically the same thing. I have replaced the regulator, the injector, did a tune up. The problem mainly occurs when sitting over night.

This post is meant for everyone who gave me advice regarding hard running/ backfiring Vortec V6:

Today, returned to jobsite with a highly experienced gas truck tech and continued diagnosis. Both of us working together, we removed cover from mixer and removed diaphragm once again.
Immediately, we heard a suction sound effect being made when diaphragm was being removed. Literally had to peel away diaphragm in order to remove it from top of mixer.
Diaphragm looked fine, reinstalled it.
Removed propane intake hose from bottom of mixer. Again, we clearly heard a suction sound effect being made as the hose was peeled off of elbow fitting.
Installed hose and turned keyswitch. Engine started right away. Idle and high rpm quality is very good. No more backfiring.

Not sure where this extra vacuum came from. Might have something to do with changing the LPG fuel filter.

The gauge cluster for this forklift looks as though it is one of the latest designs that are used by a Yale gas truck. Definitely not "idiot lights".
Thanks to everyone for their knowledge. I will put into practice all of this information when I return to work.

does the truck have a premium guage pack? or is it just idiot lights?
if there is an oil pressure guage along with a light then there may be 2 senders, one on top of the intake beside the distributor which is just the switch and then one mounted on the side of the engine near the oil filter which is the sender for the guage.
if either measures less than 2psi for more than 30 secs it will shut the engine down, so if one is defective it could be causing this.

checking the vaporizer as kevin and others suggest would also be a good thing to check, the symptoms you have lean more towards a fuel issue anyway but i just suggested the oil pressure switch/sender because they are known to cause problems.

Yes at the vaporizor........check your fuel PSI when it IS running and when it quits...

I did not recheck the spark. Do you mean the fuel pressure at the Vaporizer?
It was quitting time when the engine shutdown and would not restart.
Returning to jobsite Monday morning.

Alright. Won't be until Monday morning. If I do a resistance test for the oil pressure sensor, what ohms resistance is good and bad?

When you attempt to do the restart after it shuts down do you still have GOOD spark? have tou checked your duel PSI when it is running and when you are doing your restart?

now that we know its a vortec v6 check the oil pressure sender...
if it goes bad it will not let the engine start (i've seen these go bad frequently)

also may have a crankshaft position sensor mounted up next to the crank pulley that may be defective.

Thanks for responding.

Some information did not appear in my original post.
Let me clarify:

Make: Yale
Model: GLC120MJNGAE084 (propane)
Serial: D818V02027C
Hours: 8381
Engine: Vortec V6 with Impco Regulator/Vaporizer; Impco Lockoff not used.

I will answer your questions now:
Did not originally adjust air/fuel mixture at carburetor; since I found that Plugs and Wires were old and worn, replaced them first.
(Was able to replace Distibutor Cap and Rotor today, also replaced LPG Fuel Filter).
New Plugs are gapped correctly.
Did not find any vacuum leaks.
According to the forklift operator, backfiring seemed as though it would get worse as the engine heated up.
Was able to remove cover of carb/mixer without the screws stripping out the threads. The diaphragm looks fine. Reinstalled diaphragm/ cover.
Spark quality is very good, with new parts.

Today's results:
Engine started right away (with the new tune up parts) and remained running approximately 45 seconds without backfiring, then it shutdown.
Attempted to restart engine WITHOUT ether/ starting fluid at 3 separate times. Engine rotates, will not start. Plenty of propane in the tank.

This new problem reminds me of a bad Regulator/Vaporizer that prevented Propane from reaching mixer.

air leak on intake side can cause backfiring
vacuum leak on any hose connected to the intake side
of course cracks in the distributor cap can cause this (usually accompanied by a miss on the cylinder associated with the electrode)
bad fuel / air mixture

got serial number?
my W.A.G. = bad distributor, or 'bad pick-up' in the distributor.
you waste money by doing only part of the tune up, as the spark will want to 'go somewhere' and when it goes somewhere it should not, you ruin your new spark plug wires by not having a new cap and rotor at the same time to change them.
you DID gap the plugs didn't you?

Do the symptoms change when the engine's cold/warm? How's the ignition spark- bright blue spark if held 1/2" or so from ground?

Usually backfiring comes down to a weak mixture so first try adjusting the mixture screw on the carburettor and see if that dose any good.

Regards Nero