Well, you can't make your W do anything legally of course but in your shoes I would have your attny file an immediate motion for immediate reimbursement of legal fees. Whatever you do, do NOT agree to ANYTHING between you and your W or via the mediator about legal fees. My attny told me that every day and my H refused mediation. While this sounds funny I basically told my attny that I would go to the electric chair or spend the rest of my life in jail before I paid a dime of legal fees. Not only was my H ordered to pay my legal fees, he also had to pay my mom a penalty as she let me borrow the retainer and had to open up a retirement account to get the funds. And, it is HIGHLY advisable not to take any legal fees from joint funds as it demonstrates need that you had to borrow the money as to not take away from your quality of life.
The one beauty about living in a "fault" state is how acceptable it is for the spouse who is NOT cheating to file for divorce. It is not what you desire however it *is* considered cruel and inhumane treatment to be expected to stay in a marriage when a third party is present. You W can easily be portrayed now as a "flight risk" and it would not be unreasonable to have your attny request her passport be surrendered. Chances are she will request the same and of course, the passports of any minor children. Especially since NY is very close proximity to Canada with entry points to the country peppered in several areas. This is very common procedure.
While I know people that have made it through mediation just fine (and most of them were 2 spouses that were "on board" with a divorce) mediation is not really conducive to the divorce laws of this state.
It is *very* hard and very scary but you must be very aggressive with your desires (legal fees, support, asset allocation, division of debt and property) as there is no set formula just general guidelines so you can create some leeway for yourself if you approach it with a very skilled attny.
I would gather your marriage is classed as "vintage" (has reached the 10 year mark as mine did). Sadly, it is harder to dissolve a vintage marriage in this state and to "win" what you desire expect things to be contested unless you have a magician of a mediator.
In your case, your W contesting the grounds would actually be good for you as she would need burden of proof, not you!
I wish I could put a positive spin on all this but it really is horrible. Everybody told me that and I was so sure my case would be different. It's not.
It is SICKENING to fork over such a massive check to an attny for this reason but honestly, it's just too much for any normal human to take on themselves. While the laws are stringent they also are filled with loopholes and once you get stuck in a loophole it is NOT easy to get out!
I studied the divorce laws of this state like it was my job. I was very selective in retaining counsel and while it was a ton of work I am so, so glad I put the effort in that I did.