I would personally hang the halter on each stall door with a hook or something. You wouldn't have to run back and forth for each halter, you just run to each door. That would also help with fit. You wouldn't have to worry about which halter fits which horse as the one hanging on their door is specifically for them.
Also, like someone else already said, leather halters would be best as nylon melts at a lower temp. It would be more expensive in the beginning, but by god if it saves your horse's face from burning and them thinking they're going to die, then it would be best to shell out some extra money.
It's a great idea. Henny always has a lead rope tied around his paddock gate. It can easily/quickly be undone and clipped around his neck. He's well versed in walking around with a lead around his neck as that's how I bring him in every day. He also has a couple halters and leads hanging up on his stall door. It seems to be something a lot of people at my barn don't do. It would be a shame if their horse was lost because they couldn't find a halter or lead. :/