[rwairforce]

You wouldn't need to go over the complete lawn with the sweeper.
Has anybody cut, raked and baled hay or straw?
If you mow pointing your grass exit side of the mower towards the uncut part of your lawn and keep doing that, you will keep cutting your clippings as well as make a nice row of cut grass. You can then pickup the row of grass with your sweeper with just a few passes.

 

[rwairforce]

I think my method with the sweeper is much faster than messing with dumping the collection bins over and over while you're trying to get the grass cut. They can fill up pretty fast. Plus I think you would have a lot more cut grass to deal with.