I started playing the Fleet vs Khalians campaign and the Weasel 1 BBs seem impossibly powerful. Looking at the stats, they only have 1 more attack than Lexington 1 and 2 more than Essex 1, with comparable ray defence. Yet somehow when I attack them with my battleships, I suffer twice as many losses as they do, with an 8-ship unit of my BBs getting wiped out attacking a 6-7-ship unit of the enemy while only getting 2 kills in return. This is how it happens *every single time*. What am I missing?

You could try swamping them with cheap Blucher Destroyers whilst keeping your battleships at max range, there is some kind of hidden modifier that makes successive Destroyer attacks runs more and more damaging (presume its supposed to simulate torpedo runs), you will lose a fair proportion of the DDs to return fire, but they are meant to be expendable.
If "Supply" is turned on, the Weasels will run out of ammunition pretty fast when facing down a Destroyer hoard; then your battleships have free shots at them without the threat of return fire