I'm wondering about this weir you mentioned. Very small floating plants like duckweed are probably going to be washed over the weir. Any floaters are going to get pushed to the weir. We can't see the design, so you must decide what will and won't work.
Anyway, Hornwort, (Ceratophyllum demersum by the way), left floating, is a fast grower, so is Indian Fern/Water Sprite (Ceratopteris thalictroides), can also be left floating and would also be a good choice.
I suppose a lot of floating plants could cause gas exchange to drop, but I've never experienced that, or heard of it really.