I would not dig or divide my plants in the middle of summer. Bindweed is not impossible to weed, just another garden problem that must be solved. Al

I see why you want to start fresh with the bed; your bindweed problem sounds formidable! I would dig large rootballs--dividing them would stress them much more. Keep the rootballs moist and in the shade. If you have enough pots, that's the best way; on plastic covered with wet newspapers or burlap is an alternative. After you've gotten out as much bindweed as you can, and have re-planted everything, check the bed on a daily basis for sprouts and weed them out. Persistence is the key. I have eradicated bindweed from a property this way.

Calistoga's right--midsummer is the worst time to divide and replant (except for bearded irises), but if some of the plants you're moving are particularly hardy, you could divide them successfully if you keep them well watered, and possibly shaded on hot days.

I agree with what everyone has written - I would just add that one key to success in transplanting is maintaining a good ratio of roots to tops. If you disturb the roots severely, you may be able to save the plant by severely cutting back the tops. If these are all perennials, it probably wouldn't hurt to cut them back to the crown. That reduces transpiration thus reducing the plant's need for water while the roots are establishing after transplant. And of course the more intact the rootball, the better.