Honestly, it depends on the TV or monitor in question. Some TVs, if you leave them on and kill the power from the surge protector (or unplug them) will "remember" that they are on the next time you apply power. If you can unplug the floor model of the one at Sam's club, then plug it back in (or ask an employee to do it), you can test it out.
Alternatively, with minimal effort, you can extend the wiring from the on/off switch in the TV. You could connect to another portion of your cab to unify the "on/off" function of everything (a little more complex, but still not tough). Or you could even extend the TV's on/off switch wiring to a simple switch (or arcade pushbutton) on the outside of the cabinet if you want. My friend did the latter, deciding to have two hidden buttons (power on the cabinet, power on the screen) that he would press to get everything running. There are tons of options, especially if you're not afraid to open up the TV and do a tiny bit of wiring.
Try the unplug trick at Sam's club first if you can, worth a shot before considering the other options.