[VFR750]

Sounds like the rear axle bearing is shot. With a good bearing there should be ZERO play.

This is a duplex ball bearing, with a very strong press fit. Save yourself some real grief and have your shop do it. It'll cost about $450 to do.

If the bearing is really that loose, you will notice improved handling once it's replaced.

Unfortunately, you may want to step up to the plate and replace both sides. I say this because one side is really bad. That may be an indication of it use and exposure. Now your talking $900. That might be the price point where you decide to buy the tools and learn how to swear. My shop charged 3+ hours to do one side and claims its a PITA.

Ultimately, you have to decide how much you want to do yourself. IMHO, this was just too much learning for a once every 10 year thing.