P0420 light has been coming on and off intermittently and now it's state inspection time. Help?

I have a 2008 Prius. In August 2019, I was driving across the country when my check engine light came on. I had it read, and it was P0420. Learning that it likely wouldn't cause harm to the vehicle itself, I shrugged it off, kept driving, and decided I'd deal with it when it came time to get my state inspection.

Throughout that winter and the summer/fall of this year, a curious thing kept happening. The CE light would come on, stay for a while (a few days to a couple weeks) and then turn off again. This has happened on and off for over a year now. It seems like it may have some correlation with changes in temperature, but I can't be sure. And now, the can I kicked down the road is upon me--I need to get my MA state inspection.

Throughout the year, I've tried to find instances online of the CE light coming on and off for a P0420 code and haven't found much. Does anyone know if the intermittence of the light might indicate a particular cause? I know the catalytic converter is the most likely culprit for a P0420, but I'm wondering what the on-and-off occurrence signifies, if anything. Could it be an O2 sensor instead? Does the cat vary in its compliance with emissions standards, causing the light to come on sometimes and turn off other times? Could it jostle around and get knocked into and out of alignment?

Most crucially, what should I do for my inspection? Most recently, the CE light was off for many weeks, and then turned on last Thursday. It turned off again yesterday. My initial thought was to go get the inspection now before it turns on again, but reading up about the emissions inspection procedure, I'm not sure if that would cut it. Is there any way I can time things so I'd be likely to pass? If it stays off for a few weeks and lots of travel, would that make it more likely to pass? Would getting an oil change beforehand help? I'm nearly due for one.

If it's relevant, my battery ran out and I needed to get a jump a week ago on the night of Sunday the 6th.

Thanks in advance for any help! I know very little about cars, so I appreciate any assistance.