The comparisons to the Conventions are useful, but I think somewhat misplaced. Conventions are generally held in convention halls and sports arenas. Now, admittedly, the Roman Senate had no qualms about meeting in theaters when needed.
I'm just struggling to imagine a proper legislative chamber built to the specifications necessary to house thousands of delegates.
One thing is certain, however: The Senate would be even more prestigious in comparison. It would be almost impossible to properly debate in a chamber of 3000+ legislators, so I could imagine much more of the informal authority being ceded to the Senate; perhaps the House would tend to take its cues from the Senate.
Pity on the House Whips, too. Imagine trying to get 1000+ legislators in line. Might be interesting effects from that... If you've got a sizable majority in the lower chamber, you've got to keep hundreds, maybe even a thousand more Representatives voting the same way, than the minority party does. It could lead to a more unwieldy majority, regardless of who has it, and then, from there, cause the majority to switch back and forth more often.
And, of course, Gerrymandering would be less of an issue.