Hi R.,
I appreciate the fact that you want to have your boyfriends' approval, consent, permission (?) before you go to your Mom's house, but, really, if you want to go, and are taking steps to ensure that everything is clean and fine for your daughter when you get there - then, sweetie, just go.
10 days, in the life of a two year old, is really not that long for your BF to be away from her. I don't know what your dynamic is with your BF, but I think it is okay for parents to take their children to visit family without each other having to be present. It is also okay for him to worry about his daughter - BUT, and this is the big one - you are not under his control. You are your own person, and a Mother, and you have to make the decisions that are in your, and your daughters, best interest.
From your post, it sounds like spending this time with your Mom is important to you. Please don't let him stand in the way of this and get in between you and your family. I hate to beat the drum - but you are not married - and while you want him to be happy, you cannot forgo your own happiness to appease him. The fact that he has never let you go visit without him raises a red flag about control for me. Doesn't mean its true, it is just my reaction to that fact about the dynamics of your relationship.
Take the trip
Have fun
Enjoy your visit
Good Luck and God Bless
ADDED: R. I just read your post from earlier in February - I changed my mind - go now to your Mother's house, get an attorney, and file for full custody of your daughter. Get everything all legal and through the courts, get that job at your Grandma's store, get your diploma, and go to college, get your driver's license if you have not already done so, take charge of your life so that you can provide for your daughter without being so dependent on your BF. You deserve better.
Peace, Love and Hugs.