Surely you have the rest of the school year to try to sort this out, OP? I suppose a change in living circumstances could flag something up, and, yes, total household income will be taken into account along with things like property ownership, etc., but I've never heard of a school trying to make changes mid-year. And, in fact, unless they were alleging your partner had committed fraud, I would imagine they'd be breaching their own contract.
Generally speaking, private schools use outside services to assess and verify income and it's re-assessed on a yearly basis. It's an unfortunately long and admin-heavy and somewhat invasive process, so I would imagine there's been quite a lot of this happening before you were asked to sign any kind of agreement for next year?
With that in mind, I know others have made this suggestion, but surely the first course of action is for your DP to make an appointment to have a face to face with the head and the bursar? Generally in these types of circumstances schools are quite keen to make things work.