Refuse to pay and get help with a lawyer You did nothing wrong

I'd seek the advice of a lawyer. The people you interacted with paid no attention to what you were telling them, and you were then made to undergo examinations against your wishes. The hospital is at fault for not listening to you, and you shouldn't have to pay for something you didn't ask for.

tl;dr: Cops & hospital not to blame, caller to blame. Lawyer up and nail that ****** to the wall! Sadly I had to put my bestie/spouse in the psych ward against his will a few times. He would make up any lie he needed to in order to be released. One time the doctor even reamed me out for "dragging busy health professionals into my squabbles and wasting their time", which implied Bestie had told them I made his suicide attempt up to cause him grief. Those idiots acted like they never heard a person lie to gain freedom before- super gullible, fell for every excuse every time. Four times he was released within 48h. There were iirc six other suicide attempts where he seemed stable enough by the time I found him (ie drank pesticide after I left for work, was recovering by the time I got home) that I heeded his pleas to not rat him out. It seemed pointless to bother anyway since he never was held long enough for any benefit. He finally did die. So yeah, it's not entirely a bad thing that these cops & medical staff were disbelieving of OP's claims. "I dunno what my friend is playing at here sir, I have never been suicidal in my life, honestly!" is an excuse used by the majority of suicidal people, I'd wager. It would result in deaths to just believe it. Who's to blame here is the caller. They deserve to be charged with misuse of emergency services, and sued for the two grand plus pain and suffering. Perhaps there's other stuff a lawyer can think of too, maybe harrassment or slander or lying to police. I bet the authorities would be eager to cooperate once they understand. This is like "SWATting". It could be a huge problem if this creep who did this gets away with it and tells their creep friends, and they tell theirs, and suddenly hundreds of people are doing this to rivals or exes. Throwing the book at this creep now, nips an awful fad in the bud. Lawyers usually give a free consult so it's risk free to look into your options & odds of success.

Send the bill to the police department for being idiots.

Ok so I don't know all the laws regarding healthcare in the States but we also follow APA guidelines and crisis procedures. OP said that it was an anonymous tip but the police could just be protecting the tip maker's identity. If the police were following procedure they would have inquired about the reason the individual thought OP was suicidal and part of that is ensuring that the tip maker knows the OP well enough to make those claims (e.g. therapist, room mate, or family member instead of a random coworker who thought you were acting depressed or a shop keeper who sold you a gun and ammo). This being said can't OP foot the bill on the tip maker for wasting police/hospital resources? Even if the police for some reason didn't take down the name/number I'm sure they could trace the call back. It's the caller that has to pay the bill here if the call is deemed a waste of time not the person they sent 911/the police to, and it doesn't matter if the call was anonymous they find and fine the person. If OP can't foot the bill on the caller I would recommend using a lawyer but I know their fees can be so steep it could cost more than just paying the bill.

This, somehow, reminded me of the TV show ‘Awkward’.

This just doesn't sound right.