I don't think the type of mange that affects foxes/dogs/cats/etc. can jump to chickens or other birds; the mite species affecting birds are different. Other mammals would be at risk though, and rodents may have already gotten to the carcass and have become a vector. So if you have dogs/cats they would be the main ones at risk.
Permethrin can be used to control most mites in an area. You can either spray a liquid or use a powder. It is toxic to cats and bees (and other insects) but also degrades fairly rapidly in the environment - so if you have cats in the equation you don't have to keep them away from the area indefinitely.
If permethrin is a no-go then about the only other option would be DE. Depending on the composition of the bag, it can have some respirtory risks associted with it if used in animal enclosures. However, since it sounds like this would be outside the chicken enclosures, it would be safe to use it there. I have used DE perimeters to help control the spread other types of mites like grain mites that have seasonal population booms in my region. DE is less effective than permethrin except in rather large amounts and it gets washed away easily by rain (although the same is true of permethrin).
Because of the rodent vector risk, if you have mammal pets or other livestock to worry about then you may also want to set up some traps or refresh existing traps in the area just in case, and then use disposable or disinfectable gloves to remove anything that gets caught. I use tunnel-style traps around my coops for chicken and pet safety.