the guy at the muffler shop is 100% correct
It would be a darn shame if you replaced the catalytic converter, only to ruin it because the misfire wasn’t resolved
There are many potential causes for a misfire, such as spark, fuel, compression and so forth
Is the misfire on a particular cylinder?
And even if you happen to have a rather vague code, such as P0300, doesn’t mean the misfire isn’t primarily on one cylinder
Hook up a noid lite . . . make sure the injector(s) are being pulsed correctly
Use a spark tester to make sure you have an extremely bright blue spark
Perform compression test on the cylinder(s) in question. Post the results. Don’t assume that a particular value is okay. it might be too low for your particular engine.
As for that P0420, with the scanner, you need to look at upstream and downstream oxygen sensor readings with the engine warmed up. The upstream sensor should be fluctuating quite a bit, from below 200mv to above 800mv, with an average of 450mv. The downstream sensor should be very stable, not wildly fluctuating.
Exhaust leaks can also have an effect, as to generating a P0420 code. You want to make sure you don’t have any
Is the engine burning a lot of oil?
any previous blown cylinder head gaskets or intake gaskets?