Hi Imtheidiot. (i don't like typing that name.)
Just keep moving forward. keep your eye on the prize and keep the best interest of your child first and foremost. When a parent is an addict/alcoholic, it isn't always in the best interest of the child for there to be equal custody. Sometimes it needs to be supervised contact. I encourage you to focus on getting yourself settled and THEN worry about the paperwork.
If you are worried about him taking her from daycare, then contact the daycare and let them know you are separating and if he comes to get her they need to call you immediately. They will be sympathetic to your concerns.
It doesn't matter why he is forcing this issue about custody. It could only be manipulation and threats or it could be something far more serious. Try not to put anything in email or writing yet. Don't let him pressure you until you are safely situated somewhere else and able to think clearly about what is really best for you and your child. What's important is your childs welfare - it's only a few more days.
You are almost at the bottom. After Saturday things will improve. And you will be able to make better decisions about the whole custody arrangement. Do not allow yourself to be pressured sweetie. Custody and visitation is a huge, life-effecting decision that should not be rushed or decided under duress. Just put him off however you can. Tell him you are busy moving and you'll write the letter as soon as you can. If he wants to force an agreement, tell him to write it. That should give you some time... (you don't need to sign ANYTHING he writes.)
I used to tell myself "I can put up with anything for xxx time. Because it's only temporary. And I have a plan for my future."