Oh come on. As an actual single parent I have a much tougher financial situation than someone cohabiting. If I moved someone in I would obviously benefit from sharing bills, council tax, costs etc, making me more able to pay school fees as less likely to qualify for a bursary. It absolutely has to be done on household income - as benefits and university support is.
Why they didn’t ask you before is a good question but their mistake was before, not now!
For people asking about lodgers - yes, if you had a lodger they’d need to know about it as that would be a form of income / shared costs. Bursary investigations are very thorough and not just about your income, but your mortgage, savings, costs… they are analysing what you can actually afford to pay. OP’s partner can more afford to pay while living with OP than not.
If OP doesn’t want to pay private school fees for a step child that’s a fair enough decision but it means they may have to split up or the child lose his bursary. That’s just facts. Up to OP and partner how they feel about that and what they do. The school isn’t doing anything wrong here. Really weird threads with people thinking they are!
That said, from what you’ve said about the school’s response and capacity / popularity, it’s worth talking to them about it. They do try and keep kids once they’ve started with them. You’d be getting a bargain though.