Will everything you do for or give to your daughter have to be run by your partner first?
As she has two children (and you one?) does this mean that your daughter should get twice as much as hers, or half as much as she spends on them? I hope you can see that either way is nonsensical.
An equal partnership means financial swings and roundabouts, with both partners contributing what they can when they can, and neither taking advantage of the other. I don't think that your daughter staying put for a while is taking advantage, and TBH, I'm a bit worried that you are moving in with your girlfriend in the circumstances. Do you have any protection if things go wrong? If you are living in her house you might find that she holds all the cards.