I don't have recommendations on preschools because I live in Parkville (north of the MO river) but I can tell you when we started our older daughter in preschool and how it worked out.
I was pregnant and had our 2nd child one month after our oldest turned three. We started our oldest in preschool two weeks after our 2nd was born. I only started her in 1/2 days and she only went 2 or 3 days each week. Since I was home with the new baby, I picked her up and we all had lunch together. It gave her time to get used to school on a slower pace, but she got to get in the routine.
She had always either been with her grandparents or 1-2 days each week she'd stay with a lady she's stayed with since she was 1. It was a transition, but she did it happily and I think that's because we prepared her.
When the baby was coming, we told her she'd get to start to school when the baby got here. She was very excited about school. I think it really helped that she only went 1/2 days and got to come home to the safety of mommy and the house at lunchtime.
Since you'll be home, it may be nice to find something pt so she can transition.
Oh, I do have a referral. One of my friends used the preschool on 87th just west of Lackman in one of the huge churches (there are two together on the north side of 87th and she used the one on the east). She loved it. Her kids went 1/2 days and only 2-3 days each week.
Good luck!