A house party can be a good option to celebrate the new year in Mumbai. This can be a budget-friendly party option. You can have a good time, playing games at home. Either you may have food cooked at home or you may opt to order food from outside. This can be quite enjoyable.
Other than this I don't think I would recommend places as you told you to have a 3-year child with you. Too crowded.
Good Luck!!