Why shouldnt it stay 3/7? As long as the input equals the output nothing changes.
Only concern I had with your idea was that in the instant the hatches open, water will immediatley flow downwards OUT of your bath while the new water from the input would be still on its way to reach the bath - the amount of water in your pool would have dropped slightly depending on how far up the input hatch would be.
But nice engineering, I might try that 
No no it doesn't seem to be so simple. The outflow of water is reliant on the pressure of the water in the body, so how high it is as well as the shape of the outlet. Whereas the input has a constant 7/7 source of water so only reliant on my channel shape. I think if I want to have a higher level of water I'll have to increase the input by widening its channel width and decrease the width of the output channel.
I guess it would translate to a fine differential equation if you want to turn it into math. water level reliant on input and output, output reliant on water level and outlet shape and input reliant on water level source and input shape. Complex stuff xD
Infact for the sake of testing I started with the bath filled 7/7 and let it stabilize, afterwards start empty 0/7 or 1/7 here and then from leftovers, and let it stabilize. It always ended up at the same water level.