A quick look online suggests that lithium replacements for lead acid batteries are a "thing", so it should be fine.

I don't know anything about golf cart motor control circuits, but I would have thought even an electronic variable speed drive should be tolerant of a reasonable input over voltage. If the motor is just switched in or out by a relay (full speed on or off) then the only effect on a DC motor would be that it would run a bit faster on a higher voltage. Some of the golf cart battery suppliers suggest 14V as a terminal voltage for a lithium battery, but even that isn't too much over the 12.8V nominal of a lead acid battery.

Biggest difference in use would most likely be the discharge characteristics - a lead-acid battery would tend to slow down before grinding to a halt giving notice that the battery was getting low, where a lithium battery would perform as normal until it suddenly stopped.

As well as charging & storing a lithium battery away from anywhere or anything flammable, I understand that they cannot be charged below freezing without damage - some electric cars pre-heat the batteries (using the electricity you paid for to charge it in the first place) in cold conditions to allow the batteries to charge safely.

The fire while charging or storing thing should be taken seriously - my brother in NZ had his shed burn down due to a e-bike battery fire. If that had been in one of his previous houses with an integral garage, it would have taken the house with it.

Nigel B.