Using a tuner will probably allow you to present a good match to the radio for either 10 or 11 meters. That won't make it actually perform any better though. You need to figure out what the problem is and fix it instead of just applying a band-aid to it.
If you don't want to run ground strap, use a battery jumper cable to test with. Clamp one side onto the antenna mount and the other to something attached to the frame. You might have to scrape away a little paint on the frame side. If that solves the problem, you know that you need to run some ground strap.
If you don't want to run ground strap, use a battery jumper cable to test with. Clamp one side onto the antenna mount and the other to something attached to the frame. You might have to scrape away a little paint on the frame side. If that solves the problem, you know that you need to run some ground strap.