We have a picture on the external monitor! I simplified the circuit I had before - I removed all the op amps and shoved a transistor in instead. Then I spotted another circuit online and borrowed a couple of resistors from it. And I ended up with something just as bad as before. Then I noticed that the colours were inverted, so I fed my signal through the 74HC04 a second time, and I suddenly had a nice stable picture.

I also messed around with the CPLD some more, but it still kills everything when it's plugged in.