if its a jlg, we have/had a jlg1932, and it suffered from the instability you discuss,
a) being long compared to the forks
and
b) being heavy and having quite a far off centre of gravity in relation to the forklift.
consider:
you can get longer forks for the forklift, for the fork holes along the body, and does the forlkift comfortably lift at the distance that the centre of gravity.
b) consider the forklift being used to do the lift and if it could be swapped for one which perhaps is heavier and suited for a bigger load and longer forks, depending on other requirements.
c) as has been discussed further up, straps round it or to the anchor points at the bottom to brackets on the forklift or wrapping round to the machine to an attachment point on the forklift frame/guard (of sufficient structural strength) and so that it moves when you lift, carry and lower in relation to the forks/frame.
if you aren't confident with straps, consider chains, but i'd be concerned they would likely be considered too much of a faff to be used all the time by the lads.
do a little video with a camera on a tripod, and get it so that the procedure is simple, quick and easy to do, with a box that lives with the machine which has everything in it.
once you have procedure, video, watch together and get everyone to sign to say they'll do it after they've had opportunity to improve things and make it easy.
with reference to the batteries, we bought some from manbatt, after cleaning up connections between batteries inside the battery box, topping up where possible, with de-ionised water, and charging with tops off to see if the cells were bubbling to identify dead cells. (some are totally sealed, so you can't see the dead cells).