I have been in similar situation in the past… It’s usually cheaper for them to pay you to go away.

If I was in your shoes I would prefer to be cashed out at this point and handle the repairs myself. You can write up a document saying by paying the $800 you would find your own repairs and release them of any liability. It’s the last part that usually helps get the ink flowing and management would be happy to sign paper so they can move on and get back to business. If you are comfortable there is no issues or damage from a lack of proper seal on the air filter, then you have little to lose.

If they want to shop around and save coin on your repair, I would expect it to be done in a timely manner as well. It would not be unreasonable to ask for a rental car to minimize any possibility further damage as they seek a solution for you.

Just have a reasonable conversion with the person in charge, but don’t short change yourself or your time.