Since you are using plastic glazing , I suggest layering plastic and borax sprayed cardboard on the floor of the greenhouse, and building your raised beds on top of that.
Snake a length of corrugated spotted drainpipe through the base of each bed, and use a fan to move air through them during the warmth of the day.
This would make each bed a small Ground Air Heat Transfer battery.
You can plant directly in the
raised bed, or in containers nestled into the raised bed medium, or just put containers on top of the raised bed.
Because there won't be any snow, wind or rain inside the lean to, another layer of plastic can go over the raised bed, with minimal support needed.
Each layer of glazing is supposed to change the environment inside it to match one USDA Zone warmer than the actual location.
So inside a low tunnel located in a greenhouse on a zone 6 property the environment could be like a zone 8.
As for being a heat battery for the house, I think the lean to will be more like a buffer against low temperatures and wind.
You mentioned the house is on piles, how high up are the piles?
Does the house have insulation in the walls, ceiling and floors?