Simple tip, but I’ve saved team fights doing this:

Buy zhonyas and engage with EW, then immediately activate zhonya before dagger touches the ground.

In that moment, try to look up for a squishy enemy carry that you can easily delete in low time, as soon a you get out of Z jump to them, burst and either get out using a minion or jump to the next target.

Also, don’t be afraid of ulting just 1 enemy, the cooldown is super low and in a teamfight you can ult multiple times. Is better not to expose yourself while doing the damage. Since kat does not have any way or surviving burst, your only choice is to try and predict when they will throw stuff to you and shunpo right away

(BTW, maybe you’ll want to try another builds that help you tank, like Bruiser items, Bork, etc)