Congratulations on your forthcoming wedding op.
I think if you want both of your parents at the wedding, and your dad’s mistress is now his wife, then you have to invite everyone, and unfortunately your mum may be upset but it’s your day op, and I think you need to allow your parents to sort it out between themselves and hopefully take the higher ground. It’s not up to you to control your parents’ ongoing relationship is what I mean. They have to do that themselves. Sorry this must be incredibly stressful for you though 💐
If you think your mum may genuinely not be able to cope then you have to consider not inviting your dad and his wife and seeing them for a special dinner on another occasion.
FWIW, a friend of mine in her 60s, attended her daughter’s wedding a few years back, and it was the first time she had seen her ex dh (who had cheated on her multiple times) and his new wife, and she was surrounded with love and support from the whole family who knew the score and she was never left alone as everyone took turns to rally around her.
If anything it was her ex dh who struck rather a forlorn figure bc, despite being remarried, he had made decisions that had alienated him from family and his relationships with his daughters were just not the same and hadn’t recovered and most people thought that the loss was very much his ifyswim,