Can I still enter the Schengen area just as a transit passenger if my 90 Schengen days are gone (as a US citizen)?

This is a specific question on the 90/180 Schengen Visa Rule. I'm a US citizen almost out of my 90 Schengen days. I was reading about the bilateral agreement back, which lets US citizens stay an extra 90 days beyond the 90 Schengen days in either Denmark, France, or Poland.

https://www.nomadicmatt.com/travel-blogs/how-to-legally-stay-in-europe-for-more-than-90-days/

So I want to go to Denmark for 3 months and I will be using up all of my 90 Schengen days. Then I need to return to the US but there aren’t any affordable direct flights from Denmark. I will need to pass through a Schengen area. (I can also go through the UK but flights are more expensive.)

My question is whether I can enter the Schengen area just as a transit passenger? Do I apply for a Airport Transit Schengen Visa? I was looking into this and US citizens don’t need to apply for it usually because it is included, but I am out of Schengen days so maybe I need to apply for it in my specific case? Any info would be appreciated!

Edit (clarifying):

I’ve already spent close to 90 days in Schengen areas. I want to spend an additional 90 days in Denmark. So I will be leaving to the US around day 179.

Denmark is a Schengen country but I want to go through another Schengen country to leave to the US, because there are cheaper flights.

Edit 2:

My question is if it would be possible to apply for the Airport Transit Schengen Visa for 50 EUR and then I can be a transit passenger thru a Schengen airport (even after I used all 90 Schengen days).

Update: I called Denmark immigration services. They told me the bilateral agreement will work so they confirmed that I can stay an additional 90 days in Denmark. Also the worker I was talking to said that it should be fine if I enter the Schengen zone as a transit passenger (even if I’ve used all 90 days). I will research this further.