Is there any way to make it so only the GM moves player tokens?

It's not that I don't trust my players, I just don't want it to even be a temptation to run their characters across the map on the bathroom break. 

Not really.  There are some things you can do with the API that make running their tokens around more difficult, but even they aren't perfect.  The best option would be to go on trusting your players, or move them to an intermission screen when you can't observe them.

You could always just move the player ribbon to different map (possibly a blank one). But if you're set on staying on the same map, just go into the token's properties, and remove the player from the 'Controlled By' field. If the token is drawing from a character sheet, instead switch 'Represents Character' drop down to 'None/Generic Token', making sure the player isn't also set in the 'Controlled By' field. I'd just move them to another map though, as that's a bit of a hassle.

Yeah, thats what I meant. =D You could also move their tokens to the gm layer.

Just for fun, between games once, when the group was in the Lost Mine of Phandelver, I made a polygon surrounding them on the dynamic lighting layer (and of course had "restrict movement" checked). If you want to move map layers, and not have the reason be obvious, you could have an "intermission" map, maybe which plays some music on load.