YABU.
I have a huge cat allergy - sometimes my eyes swell up so badly the whites bulge around the iris, nou noum noum. It's a really sexy look.
I've still got three cats though. (Had to as our old flat was infested with mice)
If someone is never exposed to allergens, they will never develop tolerance or antibodies or whatever it is they are called, which could cause huge problems in later life. I used to have huge asthma attacks just being in the same room as someone who had held a cat, but my mobile mouser units have desensitised me quite well. I have to take Piriton syrup and an asthma inhaler anyway , for hayfever.
A pal of mine had a DC who was allergic to everything - but even she advised against de-allergenising the house when DD3 was diagnosed with housedustmite allergy; Pal said if you do that then DD will NEVER build up immunity and come a cropper whenever she ventured outside the house. (DDs hdm allergy has now gone)
Even the anaphylaxis nut people reckon minute amounts actually help allergies, although you can google that!
Piriton has always been non-drowsy for us (crikey, I wish it WOULD make me drowsy then I might get to sleep without a panic attack first) indeed, DD3 was prescribed huge amounts of it in hospital, it never knocked her out, it was the morphine that did that
I honestly DO know the misery of allergies, I have lived with them all my bloody life, can't go anywhere without my inhaler, piriton and nasal decongestant spray, and my triggers can change on a daily basis - but verily I tell thee this -your mum's cats will maybe actually help your DS in the long run, by desensitising him against those things you or he cannot control.
Although some drugs are evil, Piriton is so safe it can be taken in pregnancy (or so the emergency doctor said when I was having an allergy attack all over his NHSdirect line :) )