How long do you think your "shining" of the base coat will stay that way?
It will never really shine, the clearcoat is what creates the shine.
You are wasting time and effort to do so.
Here is the "fix" for the mirrors, since they are small.
Clean with a wax/grease remover, sand the clear off with 220 wet/dry and then resand with 400 or 600 wet/dry, makinf sure you DO NOT sand through the base coat, mask off and apply a rattle can of Duracolor clearcoat that any national autoparts store carries.
Or, have a body shop do the job.
Grumpy