Camber could well be an issue. Your tires might not have enough grip in the rain to decamber in a corner and aren't providing a good contact patch.

From the video it looks like it looks like you were having problems with rotation, and corner exit understeer. Is there a difference in your shock settings front/rear in the dry?

From your dry pressures, I would suggest 38fr/28(or less)rr in the wet. If that doesn't do it, try a turn or two stiff on the front shocks and full soft rear (that should lessen the exit understeer problem. Might cause an entry understeer problem though).
FWIW, my dry pressures on the Azenis are 50fr/27rr. 'Course my car weighs much more and I'm on a stock suspension. It's a little unnerving at first to go under 30 psi rear, but it's worth it.

As for the tires, this is my second season on the Azenis, with plenty of rain days. The tires are very good rain tires. The only issue is decent size standing water puddles.