Totally depends on your situation.
Even moving a tractor, they WILL mess up the grass. You end up leaving a trail of pooed-on, eaten-up, dusting-holes-scratched-into-it patches all across the lawn behind the tractor.
So probably what you need is to talk to your mom and see whether she would prefer ONE patch of ground to get thrashed into dirt (you can add sand or stuff so it doesn't look real nasty and get real muddy), or whether she would prefer a travelling parade of sublethally-messed-up ground all across the lawn.
A tractor that size is going to have to be either incredibly flimsy and lightweight, like the ones used for raising broilers, or will be moveable only with a lawn tractor (or bigger) and you won't be able to move it at all when the ground is damp and soggy. At the very least I'd suggest making the 'house' and 'pen' parts detachable, so you can move them separately. This won't solve your weight problem *much* but will make it a *bit* better.
Good luck, honestly it sounds to me (esp. with your problems with your mom) like you ought to build a permanent coop,
Pat