If you don't mind to pay around $1200-1500,try Morton's at Sheraton Hotel TST.It's a steak house with nice food and service.(careful-portion is big)
BLT Steak in Harbour City would be within your budget.
Maison Eric Kayser in Ocean terminal would be within your budget too.they do French food.
All these places are family friendly.Please make sure to book table and tell them about the special occasion and your little baby so they could arrange the seat accordingly.oh ,ask for window seat too!
Btw I think you have ordered Chicago style at BLT.the characteristic of that style is to charcoal grill the outside to keep the juicy in the meat while leaving the meat medium /rare .not all people would like that kind of style.