The issue isn't the babysitting money, really. It's that your older DS does fuck-all round the house, isn't in education and doesn't work - I'm not surprised his dad is fed up.
1 - stop mollycoddling him. He's 18. He needs to cook his own meals and do his own laundry. I don't know many 18 year olds in such a cushy position, tbh. No wonder he quit his apprenticeship - he doesn't need to work, he gets everything done for him by his mum!
2 - he needs to either go back to college or get a job. While he's getting a job, he needs to go to the job centre and sign on for jobseekers.
3 - he neeeds to contribute to the daily running of the house while he's not in work or education. That includes babysitting (he shouldn't get paid for it imo, he's an adult, not a teenager who is trying to earn extra pocket money), cooking, vacuuming, general chores, doing the food shop and whatever else needs doing on a regular basis.
I reckon if he did the above - contributed, and was either in work or education, his dad wouldn't mind paying him to babysit, but as it stands, he does bugger all at home while your youngest is in school, does housework AND babysits. Also, at 15, your younger son can't really get a job (expect a paper round) so it's reasonable to pay him for added chores. It's not reasonable to pay an adult to do chores, though!