Mike, I've got the perfect dog for you then...
Dander is not loose hairs but dead skin cells that have falled off the beast. Dust mites love pet dander almost as much as they love human dander. Which means the allergy could be to the dander protiens or to the mites and their protiens. It really does not matter so long as the MD presribes the appropriate vaccine(s) to
stop interupt the allergic response. If OTC stuff works, it probably works as well as, and a lot less expensive than, your not-yet-Obamacare health insurance.
Folks who are violently allergic know if within a matter of seconds - they start not being able to breathe any more. Anaphylaxis is not fun - especially the first time when you do not have an Epi-pen and are waiting for the rescue squad to arrive, talk with the hospital, get everybody to agree you need a shot of adrenaline followed by O2 and a ride to the ER to make sure you do not die later on.
A friend of mine just found out that even if there is not a law/rule/regulation behind it all the 24-hour pharmacies in town stock Epi-pens. They think his kid had a reaction to something the cat got into as opposed to the cat itself (even tho it's an inside cat). But the docs at the ER have decided that "better safe than sorry and get the Rx renewed in a year if she has not used it" is better than "My bad".
Hutch's situation, being cats vs grandsons, seems to be a no-brainer. Now if it were dogs I might be suggesting places where one can buy tyvek coveralls and full-face respitators just like the asbestos-abatement folks wear.
stay safe.