I don't think they are expecting a wealthy partner to cough up. It may be a routine review they do periodically for every child with a bursary to check eligibility, just like hmrc do for people on benefits, PIP etc. There will be a limited pot of bursary money (probably much smaller now due to labours lovely vat add on) and they have to make sure that goes to families who are unable to pay, rather than being unwilling to pay (like OP).
The bursary is a privilege, not an entitlement, and behaving negatively towards the school, such as refusing to provide information when requested, sending abrupt emails and issueing ultimatums is not going to endear them to this child's current situation. And maybe that is actually the plan - force the schools hand - after all if the bursary is removed and the child then has to leave the school, OP's partner will no longer be stretching to meet the fees.
Ultimately it's about priorities - does this child come first, above all arguments about who should pay what, who is responsible etc. OP hasn't stated at any point that helping with the fees would place her in a challenging financial position. So it's a choice.