I understand what CB can mean in your situation - it's in your child's best interests to have a room at your place, especially if he's there most of the time. Depending on your age, as a single adult with (theoretically) no dependants, you may only be able to claim help with your rent towards a single room in a shared property. That's a far from ideal environment for a child.
I've only read the first two pages so I might have missed this - does your ex have other children at home?
If not, and if CB is awarded to you, then you would be putting your ex in the same situation and your son would then have nowhere suitable to stay when he's at hers.
This is one of the biggest shortcomings of the benefit system - it does not account for or accommodate shared parenting.
If she has other children, and especially if she has three or more altogether, it will make a much smaller difference for her, and for your son when he's there, if she was to lose the CB for him. In this case it might be worth offering her back the face value of CB in exchange for you being the claimant and so able to access the additional benefits that CB entitles you to.
You obviously don't think much of her but she's your son's mother and you think she's good enough to have him for 3 nights a week. You have managed to co-parent for a few years, so while things have not always been amicable they have at least been stable.
It's worth at least attempting to discuss this with her because it's far easier to get CB changed if you both agree. If that fails, put in an application to have it changed. Go to CAB first and ask for help with the kind of evidence they're looking for and how to get it.
But in all of this (especially when discussing with ex) focus on what's best for your son, not on how hard done by you feel.
Stay off MRA forums
Is your limited capability for work a long term thing? The best thing you could do for your son is to aim towards getting back to work as soon as possible. It's an employee's market at the moment so if at all possible it's worth getting back out there as soon as you can.