Is this a real RO system? An RO system usually has a system of filters (usually 4-6), an expansion tank, and a bypass system. Your RO membrane is supposed to have trouble letting water through. A normal RO will oftentimes bypass as much as 4 gallons of waste water to get 1 gallon of good water.
I see you quoted the website on the optimum water temp. The entire paragraph reads: "The RO Buddie should only be connected to a cold water supply. The optimal water temperature is 77°F. Using other water temperatures will diminish the effectiveness of the filtration and can accelerate degradation of the membrane." The only products I found look like they are menat to be connected under the sink to the cold water supply. Can you link me to the product you actually have?
It may be that if manganese is your only water problem there may be a better easier to use product than an RO, which takes everything from the water--even the good stuff.