Can I extend my NON-ED Muay Thai visa without my police clearance on the last day?

You are confusing TWO completely different things.

The VISA you got from Hanoi was valid for 3 months so you could ENTER the country.

Once you entered (which you did on Oct 30th) that visa was marked used and the stamp that lets you STAY here is the small red stamp that expires the 27th of Jan (this Saturday)

Unfortunately that your school didn't tell you that you needed a police background check to get the extension is between you and the school 😕

I'd say IF you have the documentation from the school and the receipt that you did file for the background check GO to the immigration office and try for the extension this week

The worst they can say is no and that means you will run on overstay until you get the police report back and can go apply for the extension.

They will fine you 500 baht a day for every day past your current stamp expiration until the date you apply for the ED visa extension.

You don't really have much of a choice, GO try to see if you can get the extension, if they say you can't do it without the actual police report then you run on overstay