I hate to sound mean but aren't Social Services going to throw a wobbly when they work out that they're doing the washing up of the stay-at-home son (and his gf) and not the overwhelmed father?
I take it the son's behaviour is being tolerated because the family unit, as a financial entity, cannot afford for him to move out and pay his own rent. (I'll take it as read that the son making a house purchase is way out of the question). He is trapped in the family home but it sounds like his father is suffering from being trapped in there with him!
Of course there will be household tensions - it will be a case aunty telling nephew to stop 'using' his dad and to pull a finger out and sort out his own mess.
Moving your brother out for a couple of weeks respite would give him a break from the chores and improve his health situation but I'm guessing that (i) he will refuse to budge; (ii) he wouldn't want feel like he's going to impose on his sis; (iii) if he agreed and moved out for as little as a fortnight, the son will get the wrong idea that the property is now 'his', to use as he likes, causing major trouble when the respite session ends.