Charged incorrect fee for 24hr Doctor call-out - Company refusing to refund difference (informal mediation has not worked)

I am a UK resident who visited NYC over the Thanksgiving period. I hurt my elbow due to a fall when ice skating. My hotel called out a 24hr doctor from ‘Doctors 24 HS’ (offices are based in FL). The doctor arrived after 5 hours & then referred me to the NYU ER. I did not have valid travel insurance so had to pay for all medical fees myself. Before being treated by the 24 hour doctor, I agreed to the fees that were chargeable as listed on their website (http://www.doctors24hs.com/fees.html) as part of their 'new patient agreement'. In my case, I believed this to be $1650 as my treatment was classed as a "Doctor House Call: (NYC except Staten Island)". However, they have charged me $3700 for a “Doctor House Call "URGENT" - disrupting our current schedule with doctor arrival requested by patient within 60 to 120 minutes of patient call”. As the doctor failed to arrive within 60-120 minutes, I have requested the difference in price to be refunded. They have declined. What can I do next?