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Same here. It's frustrating because in the Android Issue Tracker it's listed as "fixed". Fitbit is owned by Google. You'd think they would work well together.
I have the same problem with my Versa 3. From what I've found, it has to do with Google Assistant. If you're running Android 13 and select Google Assistant as your voice assistant, your Fitbit will stop communicating with your phone via bluetooth. The workaround is don't select a voice assistant, or select Alexa since it doesn't seem to have that problem. If you're already stuck with the problem, best I can find is the only way to get Google Assistant off your Fitbit is a hard reset. This was a persistent problem throughout the beta trials of Android 13, and I had hoped it would have been resolved with the stable release, but alas it appears to persist. Frankly I'm afraid to try it again until someone says that there's either been a new release of Android 13 or the Fitbit app that's specifically resolved it.
