Sounds like your SIL isn't the only chld of your inlaws who is a spoilt little princess (or prince).
It's quite simple really.
He lives off you. This is only really possible as you have the earning power to carry him. If his planned 'contribution' (ha - ie getting mummy to do it for him) towards childcare isn't forthcoming, he needs to come up with an alternative, because you cannot afford to drop your hours and still maintain the household you do - ie him as a rather expensive pet.
Bottom line, before you drop to part time, HE should, and take on the childcare. He doesn't want to? Tough. He isn't the main earner, he isn't paying the mortgage, he doesn't get to choose here, and it's the obvious, most cost-efficient scenario, as you earn more than him.
Well, he does get to choose - but basically, if he refuses to step up and leaves you with the entire childcare problem then your most sensible, indeed your possibly only affordable solution is to kick him out.
- you would be eligible for benefits help, including council tax reduction;
- you would not have the expense of keeping him - he could take on that expense instead of childcare, and foot his CSA bill to boot;
- you would not live with the utter farce of calling this man a 'partner' when he's actually a total leech who can't even begin to provide for his child or shoulder any of the burden.
Hopefully this won't be necessary, because as you say there's more to him than his parasitical tendencies! - if that's true, you should be able to sit him down, explain the finances and he will offer either to go part time, or to take on more work in order to help finance the extra childcare which will enable you to keep your full time job.
By the way - DO NOT GO PART TIME. Financially, this could be a disaster long term.