The button alone will not let you save screenshots of all demos, but it should be possible to get alot of pictures. If you have many years of experince with 68k-assembler, it would be easy to get it in allmost any case (as you would now how to search for screen data). If you get your hardware description of your expansion port/trapdoor/whatever the name is, you can try to locate three outputs called INT0*,INT1*,INT2*. These are the interrupt lines. I´ve short-circuited these with a nail several times, and this will call the level 7-interrup (like pushing the button). Try it out, if you dare!
BTW. Keep up the good work with ADA. It´s a really nice archive. Is there any suggestions page?