N95 dust mask. Or a cartridge filter respirator like they use for painting, with a particulate N95 filter attachment in place. You could also use anything more expensive if desired (half face shield, full face shield, etc). Just be sure to keep the cartridges or dust mask dry to make sure they keep working right. There are also N95 masks with a vent in them.
Using an N95 mask is the only way I can manage to do coop and run cleanouts without negative effects on my lungs. If you're also getting contact issues, then you'd want some significant gloves - nitrile, etc. Long sleeves, etc. Might talk to an allergist to see if there's medications you can take that reduce your sensitivity to allergens. Allergy shots, over the counter medications, Fasenra, there's a lot of options to try.
You can also consider using a bunny suit - the disposable Tyvex suits that folks who do mold remediation or painting in a spray booth often wear - these cover your entire body and can be one or two piece, with booties.
When I do chickens, I don't have the contact reaction issues, just the inhalant issues. So I either give the chickens a post-mortem bath once they're hung up on a tree (a little dish soap and a hose to moisten all the feathers and clean them a bit before I skin), or when I submerge them for scalding, that also takes care of making sure there's no dander in the air throughout the rest of the process.
Or you could have someone else process, of course, and avoid the issues. Make sure you keep an Epi-pen handy if needed, those sound like some serious allergic reactions.
Good luck!
