Two more thoughts for you:
First, the supervised visitation may just work out well for you. Since your ex or one of her family members is to be there, the assumption is so that the baby can 'meet you' with the comfort of someone they knew present. Having these contacts supervised by a third party may be a saving grace.
If your ex, or her designated family member does anything to other than helpfully facilitate your becoming acquainted with your child, you will have a witness. One who will be writing EVERYTHING down. It will never be your word against someone else's. If things go poorly because of the behavior of the other adult, the supervisor can then state the visit would be more productive if they are not present in the room for the duration of the actual visit. That, more than anything, would go very far in the eyes of the court.
The other thought is this:
If you can, you may want to think about talking to another attorney. Just to get a second opinion. Something still doesn't sound right about this. I don't know where you are in Illinois, but if you can get another attorney's opinion regarding the feeling and temperature of the courts/judges in your county, it may not be a bad thing.
I just find it a little strange that without provocation, your attorney is willing to let this go this way. Have you asked your attorney specifically what potential or likely impact going along with this will have on future custody motions? Or is your attorney just trying to avoid taking a crap shoot with the courts? That is not a bad strategy, by the way, because there are no guarantees with the courts. Is your attorney trying to negotiate it all, including the eventual custody decision so that an agreed upon order is entered? That is always the better way, but only if what you are agreeing to in the interim isn't going to hurt you in the long run. Without it clearly laid out that you are going to do this for two months and then, barring some horrible report from the supervisor of the visits, you will then begin an unsupervised visitation schedule, you are taking a chance. How big of a chance, I don't know, but if I were you, I'd find out first.
I'm sorry, but I really don't know the answer to that as it is not something that is legislated. But it may be a matter of case law, and that is why I suggest a consult with another attorney if yours is not answering that question.