In due course your solicitor will lay down the basis on which you deal with the house etc. I agree, you need to press ahead with this.
As part of reaching a settlement the sloicitor will advocate mediation. This is not emotional, relationship mediation, but a mediated discussion about what is fair and what is lawful in the division of assets. Through this he will discover that no way is he entitled to half the equity if you have residency of the children.
They will encourage a discussion which leads to an agreement which would be acceptable in a court hearing.
As I understand it, if there is enough equity in the house that your share would enable you to buy a smaller house, but big enough for you and the kids, then they make suggest a settlement rather than a mesher order - and that would free you from ongoing involvement (fnancially etc) from this selfish git.
Put some energy into establishing how much you will have to play with as a single parent - Child Benefit, tax credits, child support from him etc etc.
Remember that you will also be entitled to a share of his current pensions and savings and anything else he has.
I would invade his privacy a bit more and take copies of all his policies, if he has any.
I am really sorry you are having to deal with this, but you sound strong and clear.
Relationshiops break down, sadly people fall for other people - but his behaviour towards you and his selfishness and lack of responsibility toeards his child and unborn child is shocking. You must go hell for leather to look after your interests in the face of this.
Pull together your team of close girlfriends - or find new ones, ask one to be a special friend to your toddler (childcare and attention once yu have your newborn) , and one to be your birth partner and godmother to your unborn baby - start building your new life networks now - they will be invaluable.
Good luck, Owl, and all strength to you.