hey WTBH, hope you can find some time to breathe and try and get grounded.
It's really difficult to get some perspective and make decisions when in the thick of it, and I understand that, and the spinning wheels in your mind, trying to make perfect decisions when there seems no way out and trying to protect your children.
after an incident between myself and ex, I described it to my therapist and there was a point where he had got into my car, and I couldn't remember how he had got in there, I was so frightened, the children were terrified and crying, and I still can't explain how he managed to get in the car. My therapist told me I had to find a way to de-escalate the exchanges. not because his behaviour was my fault, but because I can't make him be any way other than he is, and the safest thing for me and the children was to let him feel he has won.
If he wants the house I would now be very seriously considering leaving it and moving yourself and the girls to a rental that he cannot enter, with security cameras installed. If you can afford it at all, then it doesn't matter if you think that is part of his game-plan to get the kids, because staying in the house is not putting you in a stronger position.
as above if you get to keep this house, is it where you want your girls to grow up, would it be good for your sanity? if the girls have to be exchanged at the house and you are living elsewhere you could drive them there, let them out of the car and leave, or ask someone else to drop them off and pick them up and keep to the court order?
for the immediate changeover can you arrange someone else (not your mum, and not a friend - because he will try and succeed in charming whomever they are, and you need your resources on your side) someone from al-anon? someone entirely neutral whom you pay to stay with the girls for an hour until he comes because you have an appointment that clashes? who could act as a sitter for the changeover, and you not be there?
from my experience, lawyers are great for the legals of divorce, and you need one to defend these allegations about you, but they can only bring legal solutions to a situation - a DV center and contacts can give much greater insight and help you negotiate the what-if's and having plans in place for "if he does/says x, I will do y".
you and your daughters living a free and happy life is you winning, letting him think you are broken (before you get to the point where you are), letting him think he has won, is not the same as him actually winning.
hugs