You're essentially describing a closed system. The risks far outweigh any potential rewards.
You're essentially describing a closed system with no means to effectively deal with waste. Scientists have tried for years to develop closed systems. The biological load would have to be extremely light to come close to pulling it off. Ed could break out his abacus for us and calculate the amount of space required to minimize the risk of frog health problems. I suspect that it would be insane.
What you're descibing is not too dissimilar to running reef tanks. I spent thousands running filters, UV, and biological filters designed specifically to pull waste from water. In the end, reefers still must do water changes.
I suspect that waste loads for frog tanks are extremely heavy. We minimize the possibility of the frogs developing health issues by running flush type sytems.
Recirculating water for misting would be akin to taking showers daily with gray water.
I have no science or research to back this up. Pure speculation on my part.