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.