Thanks to all of you for your input and suggestions. I took everyone's advice and tried all three ways to see which one I liked the best. For those of you who are interested, here's what I found out.
The first time I mowed, I left the bucket on. It worked well, but I had to keep raising the bucket up high so that it wouldn't catch on my sheep fencing. Having the bucket on also made the tractor seem long when mowing. That extra length made it slightly more challenging to get around the few objects, (house, shed, woodpile, barn ...and 1 tree!) that I do have.
The next time I mowed, I removed the entire loader assembly. The tractor was much more maneuverable, but the front end was very light. My land is quite hilly, and if I raised the mower up, I had a difficult time steering without using the brakes. While mowing, this was less of an issue because the mower was resting on the ground, although the front was still very light especially on steep areas.
The loader assembly came off quickly and easily. Putting it back on was much more difficult. I had a hard time getting the pins to drop into place correctly. I was concerned about pinching the hoses and also worried that the loader would slide over and dent or scratch the hood of the tractor. No doubt I would get better at installing it with practice, but reinstalling it was a more difficult procedure than I would have liked. Removing the loader was my least favorite method of mowing, because of taking the loader off combined with having the front end so light.
The last two times I mowed, I removed only the bucket. It's so simple to take it off and reinstall that it only takes a minute or less for the process. Having the loader on the tractor gave me the extra weight that I needed to hold the front end down on the hills so that I could steer easily. Removing the bucket shortened up the overall length of the tractor making it easier to maneuver around objects. This was, by far, my favorite way to mow.
Everyone has their preferences, but for me, removing only the bucket was the best compromise between taking the entire loader off and leaving the bucket on. Overall I'm extremely happy with my rear mount mower and how that setup works on my hills. Now the mower is all put away for the winter and it's time to install the tire chains and the snowblower!