Thanks for your question. If you are a strong negotiator, then you may have a leg up on most others. BUT typically, a third party provides the most influence. In fact the phrase often used in legal and claims is "third party influence" provides the best chanced of success at negotiating a fee.
Attorneys in most cases can do a better job, but you have to pick the right attorney. Attorneys can make the hospital administrators defensive. If you want to use an attorney find one who is certified as an arbitrator.
You can negotiate the best you can and see what you get. AND then do not pay anything until you speak to the advocate. If the advocate can guarantee a lower bill, or will only take a fee if they win, then you should try that.
It is not unusual for someone to not get the deal they had hoped for, and then find an advocate or attorney to try to do a better deal.
Just be careful if you do it yourself, and intend to then try for a better deal, to not sign anything that limits your ability to use a third party to negotiate for you.