I think I'm lost too.
I would T the drinking water tank before the DI cartridge. I think it won't mix, but I know for sure you can at least get JG check valves in 3/8" (I have a couple, in fact you can just have one if you want it, let me know), they probably come in 1/4" but who knows where to find 'em.
The one last thought/comment I've got .. I played around with a 5gal pressure tank for drinking water and .. I'm not sure if you're supposed to take a bicycle pump or something to these things and fiddle with the precharge pressure, but I only get like 2 cups of water out of that thing before it loses pressure and the water becomes a drip-drip-drip. Maybe you'll have better luck, I sort of just gave up on it. But if you have the source water on a timer I wonder if it will sort of turn out the same way.
Seems to me the solenoid for shutting off the RO/DI may be faulty? If it's not really shutting off the source water?