P0420 + p0 300 chicken or the egg?

My wife has a 2011 Prius with 221K and counting. Was running perfectly fine till 2 days ago my son was driving and he said the check engine light came on started to flash and then stayed solid. He also said the engine seem to be missing. I took that is misfiring as my son is not very mechanical.
I took the codes It said p0420
Plus p300, 301, 302, 303.
Scanner also gave me the temperatures before and after the cat it's definitely hotter by about 180° in front versus the back
It is my understanding when it's hotter in the front of the cat than the back that means the cat is bad, clogged. I don't know how this would cause misfires other than back pressure?
Conversely could misfires send unburnt fuel back into the cat and that could clog it? But I got to admit that doesn't seem likely as why would I settle and get three misfires at once with no notice. I think my catalytic converter is clogged what do you guys think

 

This is me testing it. I posted this twice came back to work to see if I didn't answers and I had no post