I think I have a firm grasp on the situation now - and in my opinion there is only one reasonable course of action, and that is to sell.
You know that you cannot get a mortgage for the property completely in your name, which rules out the option of transferring the mortgage to you.
Your ex may be able to get the property transferred into her name, but this would then become her property, what would you expect to happen in this situation? Do you expect to continue to live there and pay rent? If so surely this becomes a buy to let mortgage and she'll incur a higher interest rate. She would effectively become your landlord, she would be liable for repair / upkeep on the property, and I would expect her to be able to set the rent at a fair rate, which may well be considerably higher than you are paying at the minute.
Or she gets the property transferred into her name, and then assumes residency in the property, meaning you're looking for a new place to live.
The only fair and equitable solution is to sell the property, other avenues are closed off due to your financial situation or would be unfavorable from the other parties point of view.
I know its a **** situation, but you need to face facts that realistically you cannot afford this house at the moment. Your interest only mortgage is a ticking time bomb, fighting this out through the courts is going to be a long and stressful process that you do not need hanging over your head with a young family, and it would be unfair to your ex and her ability to move on with her life to expect the current situation to continue in perpetuity.
Selling a house is not instant, there would be plenty of time to get your affairs in order and get the support you need (social housing, etc) - its not perfect and I'm sorry that you've ended up here but it seems the only reasonable solution.