I don't know anything about pigeons, but I'd use the Permethrin based spray that you have, at least for the housing.
Sounds like you may have some that are a bit more persistent? If so, consider dusting the birds and spraying the housing.
I put a little permethrin based poultry dust in a sock, then use it like a powder puff on the bird to "dispense" a fine amount of powder onto the feathers, this I work down through the feathers to the skin. Making sure to get under wings/around vents, etc. While powder/dust is dusty, I find it easier to apply than spray.
Wear a mask and gloves.
I was curious what pigeon suppliers seem to carry for lice/mites - I only checked Foy's but looks like of course Permethrin is common along with Various solutions that contain Ivermectin. So...Ivermectin for Mites, may be an option for the birds themselves, but you'll still want to treat housing with Permethrin.