Is the hydraulic reservoir in one of the loader frames? that makes it a little easier. I have a similar loader, I welded a water heater element to a pipe nipple and put that in place of the drain plug, then a small air compressor that is basically a refrigerator compressor with a steel line going to the bottom of the tank. Turn on that element and bubbler and let it run 12 hours, run the backhoe to mix the oil again, run another 12 hours or until the "steam" coming out goes down, repeat... More efficient on hotter days, or much more efficient if you drain the oil and do it in a better insulated container, like small cleaned out water heater tank. A refrigerator compressor will work great as a vacuum pump also, to suck the oil out of the reservoir so you don't have to handle it, or to lower the boiling point of water.
I haven't done the vacuum route, if you run the water vapor through the pump it complicates things, air bubbling through the oil works good enough for removing the water. Don't do this in a transmission, the water vapor will just condense in the colder steel top. Insulate the top of a steel reservoir so the water doesn't all condense and fall back in. I suspect it forms an oil foam and insulates itself somewhat inside.