You have the option of not saving the ride at the end. And the option of saving it to Zwift but not to Strava or Garmin.
In Strava, IIRC private rides don’t get included on leaderboards, so I think any PRs set on a private ride wouldn’t be recorded either (I’m not 100% on that, as I seldom mark anything private). If that’s the case, you could possibly still upload the ride to Strava and have it recognised for Fitness/Freshness and annual totals, but not recorded as a best time.
Zwift only stores your best time for 30 days anyway, so that’s not likely to be an issue.