I got to say at this point your problems are probably way over my head and i hope if an actual mechanic sees this post they feel free to chime in with any help! Your not going to hurt my feelings if you correct me!
I dont want to tell you to throw parts at this machine but if it were mine that alternator and battery would be replaced immediately. Especially since you had better success with a full or better charge. These lifts are very picky about voltage, they do crazy stuff when a constant voltage isnt being supplied. If your alternator is weak, this could most certainly cause your issues (maybe not all of them but some)
Regarding your engine swap....... If they replaced this engine with an engine out of a 40' jlg then you do not have the same size engine. You should have a Deutz F4M1011 which is a 4 cylinder engine. If you have the rod that pushes the throttle than its the adco control system. I know some of these are three position throttle setups: idle, midrange (one function), high range (drive, or two functions at once). Some of them push for mid range and pull for high range but that could only be on a cummins setup..... I dont think the 3 cylinder machines had this option. However if they used the throttle system off of the blown engine then maybe it could......? No idea.
Either way it should have at least an idle setting and high range when you engage a function.
I dont know that the throttle position would affect the basket swing, as long as your controller is telling the oil to flow it should flow.
