The new toilet I'm installing has a side to side rock(pre wax ring test) of between 1/16" and 1/8". Is the wax ring sufficient to overcome this gap? A level indicates the left side of the toilet as I face it is where the shim would need to be placed in order to stop the rocking and have the toilet remain level.
Install the wax ring pressing firmly all the way around the toilet. Use your level again. You can add wedges on one side by sliding them in gently. I would use more than one for stability. Don't slide them in too far or you will break the wax seal. Once you get them where you need them, tighten up the floor bolts. To finish, cut the excess from the wedges and caulk around the base.
There's a few ways to play this. You'll need to make sure you don't overtighten the toilet.
Use plastic shims
Plastic shims are readily available. It will level your toilet and keep it from working the bolts loose
Install waterproof flooring
My son's toilet needed a lot of shims (the flange was cast iron and tall). Not long after, we decided to install some vinyl plank flooring in his bathroom. We just put the vinyl plank down on top of the tile. I was amazed how well the toilet fit afterward.