Yeah the more you can keep away the wild birds the better. I would say the danger for mites comes more from having wild birds nesting in the area. There are mites which live mainly on the bird and mites that live mainly in the nest and then migrate once the nest is abandoned. Wild birds will...
No worries. Soap and water to clean the coop will do in a pinch especially if you have caught the problem early. You can also spray tea tree/eucalyptus essential oils mixed with water as a deterrent.
Make sure you retreat in 5-7 days to deal with any eggs that have hatched.
Depending on how many birds you have and how you want to treat the coop, go for the 500ml concentrate.
If you have a Bunnings or other hardware store near you, you can go suss out their permethrin insecticides and use that in the coop. Might save you some $$$
Pestene dust or Vetafarm Avian Insect Liquidator spray. You need to treat every bird and deep clean your coop, treat the coop and put fresh bedding in. Bag and bin the old bedding or burn it.
If you want to go the medication route, Exzolt if you can get it, and there is an off-label Ivermectin...