I have been thinking about your situation, and given your taste for less noise (smaller air pump) and need for more air and cooler res temps. My suggestion would be doing like you were thinking initially and make it a recirculate system. Instead of using another tub for your res get an insulated cooler (cheapie), the gravity feed back to the res will help cool your water and give it much needed air (gravity flow is more efficient for diffusing air than a pump). Insulation and isolation helps with higher temps in the summer too. To top it off your access to the root tub and res will be alleviated. You can also get a same size tub and a can of spray foam to insulate the root zone, just tape seal around the lips and cut in the squirt points so the foam will expand where you want it.
By all means pull that ho out and wash them roots man, she will be glad. Once they have turned like that they are already dead and rotting, give em a good wash with med pressure and don't be sad when much of that mass is gone, she has to get a clean root zone. Literally remove that hydroton wash the tap root then bathe whats left with a h202. Make sure to put clean hydroton back in too. Keep changing the res especially if your getting the rotten root diarrhea in there. Being persistent is the only way mango! That root rot can be a bitch! I saw fresh clean growth at the top of that pick so she is trying to hang in there.