If you have to bag then use one mower to do it. We sometimes find it easier and faster on certain properties to just pull out the bagger mower. We always run one of the Choppers over it first to actually make the cut and then do a quick run with the Trac-Vac to pick up some of the clippings. It works great and at the most we have 1 bucket to dump, usually less than that. I don't haul the clippings off except at 2 places IF we need to bag them. The rest of the places I dump it on site. If your running 2 mowers and bagging everything then I would think that there is a tremendous amount of clippings to deal with. You could reduce it by 1/2 just by vacumming remnants vs. actually cutting and collecting.
In the fall this is generally the way we do our leaf cleanups too, one mower mulching and the Trac-Vac collecting. Really makes the job quick. The only extra time is in the backpack work.