If I understood you correctly, it's running fine on LPG now, but on petrol it backfires when decelerating and won't idle on petrol either.
How did you rebuild the carby? Professionally or did you do it yourself?
I didn't understand what you meant by "power piston stopped lagging".
Sounds like a carby issue, but it's a very good thing to rule out all possible electrical issues first, including spark, spark plugs, leads, etc.
As mudnut suggested, it may be a good idea to check to see whether the secondary throttle valve is sticking. Put on a pair of protective glasses before checking the running engine from the top of the carby as a backfire can cook your eyes.
Backfiring usually means a lean mixture.
And the main causes of lean mixture usually are, either low fuel pressure(clogged filter, faulty fuel return valve, weak fuel pump, bad contacts to petrol solenoid or fuel tank/pump, carby issues like sticking valves, plunger, etc) or vacuum/air leaks.