Why can't she take the child with her to church on Wednesdays? If they have a Mother's Day Out program-they probably have some kind of nursery set up.

And a lot of Mother's Day Out Programs let the Mom's bring their children, also. Has she checked into this? Granted...it will not give her a "sanity day" where she can just have time for herself...but that is part of being a mom...you do what you gotta' do.
Has she talked to her church Pastor or Pastor's wife, yet? Sounds like she needs some serious counseling. May also be dealing with issues about divorce and it's place in her life and beliefs. The Pastor can go over these things with her in a non-judgemental way. May also have ideas on a financial safety net for her.
If she worked prior to the child's birth-say she has been at home 3 yrs...a lot of places will still hire her even though there is a work history gap. Perhaps an older person or a stay at home mom at the church would be willing to babysit in exchange for something...a barter type thing since money will be low.
She needs to get her ducks in a row, so to speak...and if she is going to leave or ask him to leave...be prepared for the consequences either way. Being a single parent is extremely hard...especially one of a toddler. She may be looking at doing it without child support, also. So, besides the single parent she will also be the sole bread winner.
Yes, escaping abuse or probable abuse is important...but sometimes...it requires planning so that you don't end up back in the same mess, again. KWIM?