Where is the 'proof' you have him for four hours a week more? I imagine he could say the same and unless either of you have an impartial referee timing everything, its 50:50
How do you know he is actually better off than you financially? For all you know he could be up to his eyeballs in debt.
And what is this obsession with what address DS is 'registered' at, what actual difference does it make? Most places I know hold both parents addresses, if its genuinely 50/50 residency then you can use either address to apply for schools.
If your struggling financially then apply for any benefits you think you should be entitled to and let the system work out if you are eligible.
As for this idea of putting him in nursery for a day, a) you will be worse off financially b)he will become the main carer as you will no longer be looking after your son half the time c)you wont see your son as much. And what actual benefit do you get, I cant think of a single one, unless you are using the extra time to work and earn more.
Seriously just chill, you have your son half the time, that is more than a lot of separated parents get.