Thanks OT.
How about texting her.
If it were me I'd just say something like "Thank you so much for inviting me, but I'm going through some stuff right now and I truly feel it would be better for me to not go, it's nothing about you or anything like that, I hope you have a wonderful time, I would love to catch up another time though, happy birthday"
Nice and simple.
I'm off to do a pile of dishes......baked and cooked like mad today......made savory muffins, sausage rolls, mini savory meat balls and ginger crunch slice.....for my sons school lunches. Then off to bed.
Have a nice lunch with you're dad.