I would water them real well get the new home ready perfectly sized for the root ball and then just cut the 3 gallon off and move it. The plant won't skip a beat. I prefer at a minimum 7 gallon air pots. If your nervous try just 1. Been in the dirt my whole life and never seen a plant that wasnt happier in a bigger pot. It's not like trying to move them outside in dirt where you damage roots.
If you do it like the above you will be golden. I wouldnt pull it out of the air pot, or try to flip it upside down to save the pot. If you do that you may damage some roots and slow it down. And yes the roots can overfill a pot to the point that is will slow or stop its growth in soil. I'm talking air pots, fabric pots, etc etc.
And yes I always pull my plants out after harvest and inspect the roots.