Had a minor fender bender (at fault), settling outside of insurance, what to do if they come back with an outrageously high quote?

You are in no position to determine if an auto body repair quote is too high. Either pay what is presented to you without hesitation or file a claim with your insurance and let them handle it. That's what you are paying them to do.

Go through your car insurance.

Get an estimate from a body shop of your choice.

Only an absolute fool would settle outside of insurance if the fool was insured.
You would be practically inviting someone to take advantage of you by settling outside of insurance.

What state are you in? In California if the damages paid out is under $1,000 the claim will not count against you and no points go on your DMV record. You should let your auto insurance company take care of the other party's damages. If you give them ANY money, then your auto insurance company will no longer help you. So, there are real dangers involved in your settling the claim yourself. Don't do it.

By declining to go through insurance you have already infringed the terms of your insurance. If you go to your insurance now they will ask why you didn't report the incident to start with.
If you decline to use insurance you're putting yourself in line for "a quote that does not look very official. The quote does not match the damages at all".

I'd start by looking up the repair shop on the internet to verify that its a legitimate repair shop. Then I'd call them and ask if they recently provided a quote to that person for that vehicle and see if they would verify the amount.

However, I'm not sure you have much leverage even if the person is lying. You are liable to them for their damages and if you're asking them to accept a settlement outside of the normal insurance repair process then you are at their mercy of what they will accept. They could just flat out tell you they want an extra $500 cash on top of the repair cost and your option is to pay them or go through your insurance.

If you do move forward, make sure you get a written statement with their signature saying that you've paid them, that they will not seek any further compensation and will not attempt to file against your insurance.

Thats why you go through insurance, because if you don't pay now you can go to court

Three quotes and you pick the lowest one or go through your insurance - people are snakes -

Either you pay it anyway, or you don't settle, and instead you let them submit it through insurance or take you to court.