As I understand it, ASIC bitcoin miners won't work on scrypt because they do not provide a 'proof of work' for the solved block. What if the ASIC found the solution, and then passed that solution on to the video card so that the video card can generate that necessary proof of work? In other words, couldn't they be made to work in tandem?
No. There is no distinction between "that solution" and "that necessary proof of work". They are the same thing. The proof of work is that you found the solution, proving that you did the work needed to find it. The whole scheme works because, and only because, the proof of work is inseparable from the block and transactions that are solved.
I understand that the video card has to go through the process to find the work itself. But couldn't that process happen with a little nudge in the right direction by the ASIC -- in theory? The ASIC is 'dumb' in the sense that it's getting the answer through hardware bit flips at extremely fast speeds. It gets the answer, but has no work to show for how it got it. I'm merely suggesting that it might be able to pass that result to the video card, and the video card could make some use of it.