do you even have a monitor in the cabinet the requires the use of an isolation transformer?? (regulated supply like K7000 etc) if you have a modern LCD monitor or a newer CRT with a switchmode power supply, you don't even need the transformer. the power supply your cabinet has in your photos is switchmode and does not require a transformer... neither will your fluorescent fixture.
worst case, tap the power after the switch...but before the isolation transformer and just run your monitor off the iso and everything else on the power before the iso. that way everything turns on at the same time.