Dougie,
Gaseous fuels are more susceptible to leaning out on increasing speed when using a carb designed for liquid fuels. However, these carbs are often the simplest and easiest to use with propane. The jet and needle are too small ideally for propane, however, the right kind of liquid fuel RC carb can work very well. You need a self-compensating type carb that is not of the air bleed type. One that has the needle assembly on a threaded shaft so that the needle moves away from the fixed jet on throttle opening thereby increasing the jet opening. These will work. The only problem I have found with these carbs is that the needle valve setting is usually within 2-3 turns of falling out. You may have to provide a packing or seal of sorts on the needle shaft to assure no air enters through the needle threads thereby upsetting the fuel-air ratio. This simple carb has worked well for me.
There is also a string type grass trimmer on the market built in China that runs on propane. I cannot remember the brand name, but the engine bore x stroke is about the same as yours. You might want to look at that possibility. You are also not too much smaller than some small commercial engines that have propane fuel systems available as aftermarket.
Jeff