The people above have pretty much covered all the reasons biking the historic center is not for sissies. I wouldn't do it either, and I rode some 150s on and alongside some pretty busy roads/streets when my knees were a bit (cough) younger.
Traffic in Rome is nuts. I've not biked the city but have been there a couple of times, including a long trek on the Appia Antica, so am reasonably familiar with the territory. And Laurel, of course, is more intimately acquainted with it than any of the rest of us!
If intent upon getting on two wheels, I'd think your best bet is not to try it for random sightseeing. The tours know the best routes; the ones less likely to completely freak their clientele out. You'd be riding "cold" with no idea what you could stumble into or onto. You might, however, look at some pre-determined routes, mapped by other bikers who've done them, that are out there in the web? Just doing some looking about, there's this one:
https://www.komoot.com/guide/54794/cycling-in-rome
https://www.komoot.com/guide/54793/road-cycling-routes-in-rome
There are details on each if you click on them, including cautions about sections where you'd have to walk the bike, and others where you'd have to carry it up or down stairways.
Just a thought. As an additional note, I see on one of your other posts that you're planning on visiting Pompeii (and your trip is in early August) ? If already worried about walking Rome in the heat, you're likely to have issues spending several hours (or more) on foot in the mostly shadeless scavi. 🥵
The summer heat makes anything other than a short distance unwalkable.