If you want to keep the 75% proportions accurate, you might as well use a 15" PC monitor since the viewable area is about 14.5" and if you take 14.5 divided by 19 you get about 0.76...
You should be able to get a 15" PC monitor in there, plus you could find one anywhere for dirt cheap. This would be a good choice because you could save money on the monitor solution. If you want the cabinet to look really nice, you could invest that money back into artwork and make it look really awesome. Custom artwork is rather expensive these days...
Best way to go about it is to find a monitor, then build the cabinet around it.

Again, I say, it is your project, and the design choices you make come down to your own (or your daughter's) personal preference.
If you feel comfortable spending $189 for an LCD monitor, then that's your right. If you feel comfortable spending $10 for a nice old CRT PC monitor from the thrift store, then that is your right also. If you want to get a Happs monitor and ArcadeVGA card and have the time to tinker with the hardware to get the original arcade accurate resolutions, then tinker away to your heart's content.
One last thing, then I will get off my

...
Try not to over-analyze things to the point where it hinders progress on your project. It can be very confusing, and sometimes frustrating to decide on stuff since there are so many design choices to pick from. I would say explore your options, make your choices, then roll up your sleeves and start working... and have fun... that's what the whole BYOAC experience is all about.
