Oh twiglet, I feel so sorry for you. I suffer from terrible anxiety issues when I'm flying anywhere, I am totally stressed out for a while before I go but, in all honesty, the reality of the actual trip is never as bad as my imagination makes it, and you would probablyfind the same thing. DH tends to be stressed when he gets in the plane but has a lot of tried & trusted self help routines and a tin of rescue remedy so,between us, we conquer our fears and I'm typing this from a sunbed in Barbados. There are lots of self help things online and, as others have said, a GP visit woukdnt go amiss.
As regards whether or not you should accept the invitation, that's more tricky. I'm sure if you don't go, you'll always wish you had and, in the light of my saying that it's possible to "big up" experiences out of all proportion,maybe you should go.
We have 2 dear rescue cats, the apples of our eyes. We live in a very rural area and they love roaming around, there's no way we'd put them in a cattery. We use Mindahome and House Sitters UK , both of which have provided us with some super people who live in and look after our cats & home whilst we're away. I definitely recommend you explore this option.