Sound guy for over a decade here [1], and I am off to shoot another feature starting Saturday. This is emphatically not true. You record as much as humanly possible on set. Even if you layer and resync it afterwards, doing all that in post is crazy expensive. On your limited budget it's self-defeating. Engineering the room response of the real location in post is very time consuming compared to capturing your raw material on location.
I'm open to helping with this project (EDIT: but won't be in the Bay area or online much until after 12/18); it pains me to see someone tying their hands unnecessarily.