It would be like having a 20’ long section of 12” pipe coming into the house. Sure it has 100 gallons of water in it when full, but there is nothing to push the water to the house. In a pressurized system, air works like a spring to push the water from the tank to the house. But the tank has to have about 75% air to do this. That is why your 44gallon tank only holds 12 gallons of water.
You could make it work by adding a compressor to the “storage” tank. You would have to run the compressor on a lower setting pressure switch than the well pump. You would also need to work out the vent/vac system that would allow the “storage” tank to fill (vent), and then close off when the compressor is pushing the air out of the tank. Then you need level controls for the pump filling the storage tank, and a Dry Well protector/timer like the Cycle Sensor, etc., etc.. Not really a feasible option as you can see.
The best you can do is to “pre-charge” the non-bladder style tank with an air compressor, and widen the bandwidth of the on/off of the pressure switch as much as you can. Sounds funny but, the more air in the tank, the more water it can push to the house. I would pre-charge the bladderless tank to about 28 PSI with air, then set my pressure switch to turn the pump on at 60 and off at 70, if the pump can build that much pressure. That would give you the most “draw-down” from a pressurized tank. That should get you about 50 gallons of pressurized water from a 100gallon size tank, as the pressure decreases from 70 PSI to 30 PSI. The 60/70 pressure switch setting would help keep the tank “topped off” as much as possible. But with such a low producing well, the pressure will continue to drop to 30 or so as a Cycle Senor or some kind of Dry Well relay turns the pump on and off according to how much water is in the well.
Also the lower the pressure you can live with in the house, the more water the tank will actually hold. In other words it would store more water working from 10 to 50 PSI (using a 40/50 switch), than it would working from 30 to 70 PSI (with the 60/70 switch). But you could have pressure as low as 10 PSI in the house with the lower setting.
Are you sure you don’t have room for a vented storage tank and a booster pump?