Have a look and see hun, 'cos it could be something called blephritis, (lids red and looks a bit like eczema), if so this could be causing the conjunctivitis, if so a basic lid care may help. This involves cleaning top and bottom kids with cotton bud and baby shampoo.
It may also be worthwhile gently massaging the side of nose just under the corner of his eye 'cos if his tear duct is slightly blocked then this again will cause recurrent conjunctivitis.
If it is happening regularly it could just be one of those things, but if it were me I would be asking for referral to eye specialist (ophthalmologist).
If GP is just chucking antibiotics at it and not looking at the eyes in detail, it maybe worth seeing if your local optician is prepared to have a look. It maybe easier said than done 'cos he is so little, but maybe worth a try.
Would offer myself but u a bit far away lol.