You need to LOAD TEST the battery to see if it is the issue....if it can't handle a loaded test then replace it.....ALSO Check your fan/Alternator belt for tightness and make sure it's not slipping on the pulley. Your charging system should read 14 + Volts at the battery while the truck is running,would also be a good idea to see what the amp output of the alternator is with a amp clamp on the battery cable,I have seen trucks charging ok at 14+ volts but only putting out about 5 amps or less under a load.

You need to test the battery & charge circuits before just throwing in a battery. Your lift has computers which may be damaged if the base problem isn't diagnosed & handled.