We had bittersweet in one of our paddocks this year, so I am very sympathetic. We usually would not have mowed that area until September to provide habitat for birds and insects, but we mowed it and that made a big difference. Then we manually removed the ones that survived repeated mowing with this awesome Fiskars weeder
http://tghyp.com/?p=947. Other good and easy strategies include boiling
water, horticultural vinegar (more concentrated than table vinegar), or a combination of vinegar and clove oil.
We also had a gaggle of bittersweet and porcelainberry creeping over the
fence from a neighbor. I used three gallons of horticultural vinegar on these weeds and they didn't mind at all. In the end my husband cut them back and then we treated the stumps. They are much more vulnerable when they don't have leaves providing
energy. This is also a good idea in slopes near streams where pulling out the weeds would cause erosion.
Another strategy is to suffocate the invasive plant with a heavy tarp, but that won't work in the middle of a lawn, it will kill took much lawn. It can work really well on large areas of phragmites, though.