Looks like you got the mirror to where you want it
.
Though most just stick with detergent and water, I gave up on that as I could not get rid of all the marks - including blue haze and other, likely non water soluble marks.
Now acetone is my best friend
. I suspect that once you have carefully removed anything gritty, friction between fingers (or cotton wool) and the mirror surface is unlikely to damage it. My eleven year old 10" Dob mirror remains unscratched.