Cash bars here are the norm from NE. I would still put it invite though.
I've lived in the NE for 40 years. I've never been to a wedding with a cash bar.
Have an open bar or a limited bar (e.g. cocktail hour or wine on the tables) or no bar at all, but not a cash bar.
If his family is from the northeast they will most likely be expecting an open bar. Here in ny, I have never ever been to a wedding without one. I think open bar during cocktail hour only will ruin your event. The drinkers will plant themselves at the bar and pond drinks trying to load up for free. I would do open bar all night, cash bar all night, or you can do a champagne toast and wine with dinner.
I personally think if people want to drink they will, either bring it with them, or-party etc. from my background, it's open bar or nothing, but that is what I am used to.
In the OP's situation I would follow what the bride and groom want and if MOG wants an open bar and insists on it then she can foot the bill. When I got married my MIL gave me her invite list that had over 100 people on it. My max for both sides, including our friends, was 125. I told her that she has to get her list down to 75 people (and that was being generous) and anything over that she would need to pay for. It was amazing how many of her "friends" did not need to be invited.