"Charlie" should have tested the o2 sensor, not just replaced it. It either works or it doesn't. "Charlie" didn't do the timing, it's not adjustable. ??
You likely need a cat. You should really spend the $120 to get the diagnostic done to find out what you need. The labour $ counts towards the $450 limit. If you did an O2 sensor, that also counts towards it but you might have a hard time since you did it in the wrong order.
The shop needs to show that replacing the O2 sensor and the cat would make an improvement. To do that, they drill a small hole in the exhaust before the cat, and test pre-post. Then in the estimate to do the cat takes you over $450 dollars you get a conditional pass. The $120 and the O2 sensor, including labour, all count towards exceeding the $450 limit.
I got a conditional pass with a new O2 sensor for $340 bucks, including an oil change and the test / retest. The total emissions work was $270 and getting a cat would have been over $200 more with labour, which is over the $450, so I didn't have to get it done.
This year, anyway.