By process of elimination let's assume it's not rain because the carpet is wet even if it hasn't rained. Let's look at what sources of water are in the vehicle:
-Coolant/water mix. I know you said it doesn't smell/taste like coolant but just for a moment consider that the coolant is piped into the car, connected to the heater matrix and back out through the bulkhead again. Have you popped off the glovebox and/or the small shelf under the steering column to get better access?
-Washer fluid/water mix. Doesn't come into the cabin at any point. Or does it? I'm trying to remember the route it follows from engine bay to the rear. Does it run along the passenger's side under the door edge trim? Those trims are easy to pull up and check. This would at least make sense (if the pipe runs there) because it would be lower than the fluid reservoir, therefore gravity would pull water out even if you aren't activating the pump.
- A/C system. Produces water when cooling the car but not a huge amount. Is it possible that there is water trapped in the system and leaking? Does the A/C blow hot and cold ok?
- External sources - Unless you have a 5 gallon water bottle in the back seat with a leak I can't see it being something not pumbed into the car.
Am I missing anything? If we remove rain from the equation surely it has to be coolant or washer fluid?
At this point mate I think I would consider draining the coolant and also the washer fluid. If there is still water appearing overnight I would set up a camera to see if someone's ******* in it every night
If it's dry I would refill one or the other and wait another night. Removing all potential sources of non-oil based liquid would be a good next step in my humble opinion.
Taking out the carpet would be a pain in the *** but you've already put in so much effort that you could waste a hell of a lot more. Check my threads for my recaro retrofitting guide. That should show you just about everything for removing the seats and trims.