I'm sorry that you feel that you can't offer these cats what they need.
Whether or not you have cats, I think you're becoming aware it is not reasonable for your son to act how he currently is. I wish I could help with that, but I can only recommend a harsher punishment. Maybe, bed early, no play time, no tv etc. Until he learns? Reward good behaviour too especially with the cats. Perhaps encourage him to feed / brush them?
I have asthma and have allergies to fur (and the oils it produces). I have prescription antihistamines, nasal spray, eye drops and a relieving inhaler. Sometimes, i will only need one item, others i will use all.
Generally I have found that my body adjusts around animals and I become "less allergic" over time with the idd flare up which is manageable. Perhaps with the correct medication, this will not be an issue?
I vacuum once a week, unless i'm having visitors. But sweep kitchen (where they eat) and bathroom (where their tray is) everyday. It's frustrating, but becomes easier if you stay on top of it and it becomes a routine.
We adopted two cats from the Cat Protection League in September last year. Many of the cats there were in distress, it was sad to see. I wanted to bring home more. 
We ended up with a 10 week old black kitten (Mozzie) and a 5.5 year old ginger cat (Gilbert). 
Mozzie was born there, so relatively un-stressed as he knew nothing else, although he was the "runt" and his mother an sibling had all been adopted already. Gilbert on the other hand is very psychologically damaged. He is nervous all of the time, the cat basket REALLY upsets him more than it would usually! This makes trips to the vet / cattery really traumatic for him. He was taken to the CPL twice, why, I will never understand as he i the most gentle cat I know.
Something else to bare in mind, is their breed and colour. According to the CPL, ginger and black cats are least desirable. Also, rehoming twi together is less likely than rehoming one on it's own. 
You have a lot to consider, but please do just that - consider all of your options and only use a shelter as a last resort.
I have nothing else to add, (but will be watching with interest) so I shall
Leave you a pic of my boys from only last night 