1 Corinthians 6 tells us not to take other believers to court, but rather to try and resolve such matters within the congregation. It says nothing about whether we should take non-believers to court, nor about what we should do if the other party refused to take part in an in-congregation process. It does not prohibit using the court system, just encourages us to use a more mediatory approach instead.
If the medical professional you are considering suing is a believer, approach them first and see what you can work out. Otherwise this scripture does not prohibit you from taking them to court.
Bear in mind though that, from the outside, the litigious nature of US culture looks strange. Americans appear to sue each other far more than people in most other countries - though this could be a mistaken impression based on what aspects of US society are actually shown in international media. And it is understandable to a degree in the medical field given the private nature of US healthcare. My only point in this paragraph is to carefully consider the practicalities, cost and likely outcomes, don't just instinctively jump to sue. You are already cautiously considering it from a Biblical perspective, and that is good. But if you conclude it is Biblically correct, make sure you also consider whether it is practically correct and don't just jump straight from "it's permissible" to "therefore I will do it". Your time and resources may be better placed in immediately seeking a cure outside the system, rather than seeking money from the system with which you hope to eventually pay for a cure by the system.