Thank you for your reply!
The thermocycler seems like a good theory, although I have run this reaction multiple times before with different genes, and those amplified well in the same cyclers. But it could simply be that they're getting old and not working properly, though my PI doesn't think that that is the problem, so I need to look for alternate explanations.
We have a robot mixer, so I definitely don't mix these hundreds of reactions manually - that would take all day!
Which is why using tubes is infeasible, because the robot is designed to work with plates.
Thank you, again, for your feedback!