But why should he? What if he were to say to you "Well, you can move out, I'll stay here and DD can split the time between us 50/50. That way she'll still be in 'her' home, etc, etc half the time. Then later on when I decide to sell the house, you'll get your share". Would you go for that? After all, if it's your 1st priority to keep DD in her home, it shouldn't matter which parent is living in said home and which parent is renting, all things being equal.
But seriously, your choices are; he buys you out, you buy him out, or the house gets sold and equity split. There is no other option unless he agrees to one, and I highly doubt he will. He isn't going to want to rent when his share of the equity may qualify him to buy a house or a flat.
See a solicitor by all means, it's always best to get your questions answered by the experts. It allows you to set aside impractical ideas and move forward with realistic expectations.
As far as what do you do once the house goes on the market if you have to live together til it's sold? You tell him that you are effectively 'separated'. That you will no longer be responsible for his 'upkeep'. No more cooking for him, cleaning for him, etc. And that you will separate finances (if they're joint now) and pay your pro rata share of the household expenses. You set up a 'child arrangements' schedule in which you set days for each of you to be the 'primary parent'. And you move into the spare room, the sofa, or start bunking in with DD. And you ignore him after that. You form your own 'new' life, go out with friends, visit relatives, do as you please.
I believe you can apply for certain benefits once you have separated, even if living in the same house. I'm not sure if the same applies to CMS since he'd be living in the same house as DD.