I've moved my 10-gallon petsmart tank by hand with about 30% water & full of substrate/etc. without any casualty.
Just be careful about avoiding any sort of twisting and you'll be fine. Having someone help you will make it go smoother.
If you really want to be cautious about it, use a couple of 2x4 boards underneath the sides of the tank and lift the boards with a partner (sounds like you could slide the tank so that it sits centered on a single milk crate with the edges hanging off, that would let you get the boards underneath to help lift it up onto the new stand).
I realize you don't want to stress your fish out, but moving the tank will stress them anyway, plus you run a tiny risk of sloshing them into the wall of the tank or a rock tipping over and crushing them, etc... putting them in a bucket of siphoned tank water for the short time it takes you to move everything with the water level down to the bottom might be more pragmatic.