I’m sorry you’re in this complex situation. There’s not much additional info in order to give a fuller picture of matters practically speaking (eg. your salary, mother’s home set up/salary, DSD’s living arrangements etc). You say the three of you discussed and agreed to this, so you would be very unreasonable to now throw the whole thing on the scrap heap because quite foreseeable financial difficulties are arising. This also suggests that you’ve been able to work together effectively in the past and it would be a great shame for all of you and especially your DSD if that were to be lost.
Agree with other posters that financially you need to tackle this as a threesome as well - ie. mother needs to recognise that father is essentially providing full time childcare enabling her to work (I assume - difficult to tell from your posts) and make a more significant contribution.
However, (and I imagine that this won’t be a popular point of view), you might also recognise that your partner’s change of situation could now enable you to seek a role at a higher salary with him picking up childcare after school for all three children, doing all drop offs pick ups etc. Is that feasible?
I would tread very carefully and consider offering whatever support you can to facilitate matters with your whole family unit and your own children’s best interests in mind. Your partner clearly wants to step up for his son (not every parent would) and, while there are sacrifices, there are also valuable lessons here for your shared children about how to be there for a family member/working together/who their dad is. Plus the benefit of having a parent around at home full time/help with homework etc.
You also need to manage your own feelings around claiming benefits - it might be an admirable view in general, but there are additional needs and the well-being of multiple people to consider so you need to be able to flex on your viewpoint here. This includes looking at activities for the other kids that will give bursary/free places, using charities which will support with food/clothing, checking entitlements to winter fuel/free school meals etc etc
If your partner has been willing to take this step for his son, it doesn’t seem to me very far fetched to think that you withdrawing your support for an arrangement you agreed, could have a very serious impact on the viability of your relationship long term. How would that impact you and your children, as well as your partner and his eldest child? What happens to your DSD right now could well also impact his adult life quite deeply (educational/vocational attainment, employability, relationships etc). What impact will it have on you and your partner and children if he doesn’t receive the support he needs and it results in him needing greater support for far longer? How would you feel if this was one of your own children?
The situation you are in is far from straightforward and there isn’t a clear solution. Sometimes that’s just how life is. It is not your partner’s “fault”. You are in this situation by mutual agreement and it is not his problem to solve alone. Things may also change over time. Perhaps part of your H’s time should be spent (if not already) ensuring that all possible avenues for support are followed, applications for special schools are submitted and followed up regularly (if available - you don’t say whether there are special schools near you).
Essentially I don’t think you can just pull the rug, but of course everyone in your household needs to be fed and clothed. Try and approach any conversation from the point of view of being on the same team and offer whatever additional support you can, in the first instance, to help facilitate the current situation. Recognise the upside of who your partner is and what he is trying to achieve, even though the situation is not ideal. You are more likely to be met halfway with this approach.
PS Bear in mind that there may be a genetic component to this - have you considered whether either of your shared children may also have additional needs? It would be ironic for you to create an impasse on your relationship over this only to later discover that another child would also benefit from additional parental input.