Check the ph in your new place to see how different it is from your old water (obviously test their tank water too)... drastic changes in ph can lead to ph shock. I wouldn't even mess with the airstone if it's only 15 minutes away, they will have plenty of oxygen.
I would:
1. Catch all fish right before the move and put them in the rubbermaid, move them directly to your heated car.
2. Drain tank, keep sump wet, move that and your other tank stuff to car.
3. Drive over to new place, unload fish in bin into house, check on them.
4. Setup tank, fill with water as close to temp you want to keep it at, adding your dechlorinator (sounds like you're using amquel, that's fine). Amquel (and anything else) dechlorinates in a couple minutes at most. At this point you have to decide if you need to acclimate them to the new ph. Your beneficial bacteria is mostly in your gravel/sand, and in your sump, not the water column, by the way.
5. If necessary, acclimate, either by drip acclimating or dumping small amounts of water from your new treated tap water in to the rubbermaid.
6. When rubbermaid water reads the same ph as the tank water, net the fish out and put them in the tank.
Done.