Herrick:
I tried a 15' Batwing mower for a week on my 3720.

The dealer had a used one come in and dropped it off for me to try. There is a post on here with pictures:
http://www.tractorbynet.com/forums/john-deere-owning-operating/103614-my-3720-15-wing-mower.html
With 35pto HP I didn't suffer for HP much and (you'll have to read the post) but probably mowed at 5mph. I have a single short steep (maybe a 10' rise) hill but that's it so I can't give a review of running it on hills. I also was mowing I think 2 weeks growth.
It weighed 3200lbs which was over 1/2 the fighting weight of my 3720 cab (about 6K [loader frame, loaded tires, etc.]). I would imagine that it could wrestle the tractor around a bit on hills but in my short use I didn't have any issues.
It had a drawbar hitch rather than a 3pt which I thought made it handle better. Technically the drawbar weight for transport (wings up and off the ground) exceeded the max recommended for the 3720 but the operating tongue weight was fine.