OK, first let's look at this from your partner's and his ex's point of view to keep it simple.
Mortgage|€303,000|2.68% - 30 years left
= Adjusted mortgage|250,000|4.35%
Home loan|€35,000|4.69%
Total|€285,000
Value|€250,000
Negative equity|€35,000
A tracker mortgage has a real value. If so, his repayments are around €1,200 a month. A loan with a standard variable rate of 4.35% has repayments of €1,200 as well. So in a sense, the real value of your partner's mortgage is €250.000
So the negative equity is really around €35,000
She should pay your partner around €17,500 to exit from the mortgage.
Now from your point of view...
You would be taking on €90,000 of negative equity.
If you split up with your partner, you would be on the hook for the full mortgage as your mortgage would be joint and several.
Have you higher earnings than he has? You are in a better personal position, as you don't have maintenance to pay, so the banks might be more likely to go after you.
So you should not buy half of the house and take over the full mortgage.
I don't think that you should do anything until the negative equity is removed
You can pay your partner rent and he can use it to pay the mortgage.
What is wrong with just paying the interest on the mortgage for the time-being? That is the perfect solution for a situation like this where the ex is not paying. If you or your partner pay capital as well as interest, you will be reducing the negative equity and his ex will benefit from this.
In theory, after a few years, if house price increases and capital repayments return the house to positive equity, she could claim half the equity.
Your partner should seek a payment of €17,500 from her to allow her take her name off the mortgage.
She and her new partner probably also want to bring this to an end. If they can't pay the €17,500 then they can't finalise matters.
In the meantime, your partner should pay only the interest on the mortgage. I doubt if the bank will seek to repossess it while they are getting interest.
Brendan