You are apparently happy with the arrangement and would take the children full time if you could yet you state in relation to the eldest 'obviously, we can't keep looking after the child, financing her, and being responsible for her, if we have no legal right to.'
Are you suggesting that if you can't sort out a 'legal right' in respect of this child (which clearly you won't, as neither of you are biologically related to her), that you will stop having her live with you and her two siblings for half the week? Or that, if the children's mother doesn't give you some of her SW benefits (which are, after all, allocated to her by SW, not to you or your partner), you will try to fight this through the courts?
I would imagine that, theoretically, your partner should be entitled to part of the child benefit received for the two children that are his - let's say to half (although that really depends who pays school, health, clothing costs et al). That is 300 a month divided in half so 150
For 150 a month, you are prepared to enter a protracted legal battle with your partner's X? Using a solicitor (whose fees I imagine would be at least that per hour - plus VAT of course)? Potentially causing huge disruption, unhappiness and division to the children involved?
Why? Why? Why?
If your partner has his children's interests at heart, I can not imagine why he would do this.