When I feed it, it's not a worry because they leave not a shred, but if you're concerned, you can just strip off leaves and fed those.
Yes, it is a VERY invasive plant and easily roots. We just pull up new unwanted growth and feed it to the buns. We keep a small intentional patch of it growing just for them. As an aside, there is a product called Pronto that does a very good job at clearing it out. It works on the roots, so it does take a couple of weeks to see results, but we use it to keep a "fire lane" between the back edge of our place and the kudzu.