I've been thinking about this back and forth for a bit and I've finally come up with a little theory that I believe might explain what's going on.
LPG has less energy than gasoline and (if I got the right numbers) it also has a higher viscosity than gasoline.
Now my theory is that this would mean the fuel injectors not only need to inject more LPG (than they would have to inject gasoline) into the cylinders but the LPG also has more resistance/friction due to the higher viscosity (to explain it in simple words: the "consistency" of the LPG is a tiny tiny tiny bit more like honey than gasoline is and therefore flows a tiny little bit less easily).
So I'm thinking maybe the fuel injectors could just be a little bit clogged up, to the point where they still operate well enough for the engine to run on gasoline but cause trouble and misfires whenever it runs on LPG.
I'm just guessing here, that LPG is somewhat more demanding on the fuel injectors to work 'just perfect' in order for the engine to run right than the gasoline is.
It might of course also be that the fuel pressure of the LPG delivery system isn't quite sufficient or whatever else.
So as a result of my thinking:
I would simply suggest to just go and try a bottle of fuel injector cleaner/treatment and pour that it into the (full gasoline) tank (then drive around with the car only running on gasoline until the tank is pretty much empty and the cleaner has finished doing it's job) - and then afterwards just try and see if that helps anything on the LPG side of running things. (And maybe try again with another product from a different brand if the first didn't help.)
Just to give you an example of the fuel injector cleaner I'm talking about, it's something like this bottle here for example:
IC5 Fuel Injector Cleaner and Upper Cylinder Lubricant | Sea Foam (seafoamworks.com). There are a bunch of different brands that have similar ones.
Edit:
My old OHV engine for example runs 'perfectly fine' ever since I replaced the fuel pump. But it's just old and so it runs on the 'lean side of things', meaning the fuel trim from the ECM is so high that it's pretty much just below the threshold of throwing a check engine light most of the time. That's not really uncommon for older engines that run with with older sensors and old fuel injectors and (although I couldn't find any) possibly some tiny vacuum leak(s). So it's absolutely nothing I worry much about as long as the ECM can still regulate it sufficiently with the fuel trim.
But although my old engine still runs absolutely fine on gasoline, I would indeed imagine it might currently not be running just good enough to still be able to run on LPG, if it were a converted engine and was otherwise in the same condition it is in.
About the sway bars... if it's not the 4WD issue that CdW6212R talks about (which I find the most likely explanation), then it might even just be a pair of broken sway bar links.
Those are cheap and super easy to replace, should they be broken. They just wear out with time, especially when they were tightened down excessively when installed. They cost something like 20 $ (US dollars with the $ sign right side up) a pair at the common auto parts stores and can easily and quickly be installed within just a couple of minutes. A professional mechanic is going charge whatever they charge to roll up their sleeves to do it of course. You can usually see if they're completely torn just by looking at them.
There are plenty of threads here in the forum with instructions and pictures.
I'm gonna go buy a camcorder as this will be a mystery rebuild project to entertain me over the next few weeks. When I'm out I'll buy new link bushes calipers as it's undriveable without those fixed.. George the Russian only charges $50ph...bargain mechanic for these parts.
