Even KBB / NADA are sometimes not representative as a market (although you have to start somewhere). If you told me I could find a nice 2004 Honda coupe for $5400 at a dealer around here, I'd laugh. Not in a million years. On the flip side, I couldn't get $3k out of my reliable, paid for beater, so why get rid of it? I'd say for you, if you really don't need a car, and since you are in Chicago, it wouldn't hurt to get rid of it, but it also wouldn't lose much value if you had a true need for it and wanted to hold on to it (registration / insurance / etc. aside).
As far as the AC goes, if it is fall / winter in Chicago when/if you go to sell it, I wouldn't fix it. If it's spring / summer, it will be hard to unload at a decent price with busted AC.
But here's my main concern - you said your fiance will lose a bunch of money if she sells her car. That's true. But she'll lose even more if she holds on to it, makes every payment, rides it in to the ground or tries to sell it later, etc. Let's say if she was able to sell it now private party she would lose $4k. Ouch. But if she pays it off, we are talking 5 digits of loss. It sounds like, all things being equal, you guys should be sharing that nice, reliable, paid for Honda (making some assumptions here) and just eating the cost of a lesson. Now if that Honda has 300k miles and she has to drive 100 miles a day for work , etc etc, that might not be workable. But just food for thought. I did something similar in 2005. I bought a new Focus (~12,500), tax, title, license. Then I decided I needed a house. Well I got lucky and ended up selling it to someone for that same $12,500 after I put 5k on it. So my lesson only cost me 1-1.5k. It's possible you are selling yourself short and maybe could get more than you think.