I'm in a similar situation, but I've been here much longer. Canadian embassy told me I need to be there in person in Bangkok, there is no way around it. You need to fill affidavit then have it translated to Thai then certified and it has to be done in Bangkok.
Overstaying for 2 days is not a problem. You will pay 1k baht upon departure with a red stamp which will not prevent you from coming back. I have had at least 5 of these stamps since 2010 and it's never an issue. Getting raided on 2 days overstay is extremely rare, I have never heard of it. There are thousands of people who choose the few days overstay instead of the annoying trip for 1.9k baht and if it was an issue you would heard stories here which I have never read. The fear mongering is from people who try to patronize you, do not listen to such people.
Just to end on this topic, Thailand is a country where you can negotiate almost everything. if you can bargain at hospital imagine where else you can, lol. Most people who commented lack experience and use google as their source. I knew your story would end well if you stood your ground. I mean, you can literally get away with murder if you have 500k to 1M baht to pay the family of the victim. I am not joking.
You did the right thing. Years ago I crossed Cambodia (Poipet) by land with my 5 dogs and they try to charge me $600 for bogus health test for my dogs. So your $600 reminded me of that. I stood my ground for an hour and involved the Thai side of the border. At one point they said fuck it and let me pass. I was with my Thai GF which helped, but I was not gonna pay that without a fight.
If you have been here for so long how come you were dead wrong on this one? You are giving false information to OP who is in a stressful situation. You said "The police will definitely be involved", you were factually incorrect. The police almost never get involved in such cases because hospitals are "bluffing" and in there to extract money from foreigners who are too tired to stand their ground. OP won a tough battle without paying what was ask, you were wrong. Speak from experience please and not what you read somewhere.
Factually incorrect, OP stood his ground and won. The cops never got involved even after the hospital threatened to do so and OP said `go ahead``. Stop getting your info from google.