Identify/substitute an SMD MOSFET labeled “540N 88KFG” for LCD monitor repair

Check reverse leakage in datasheet. I would not use a mosfet for "true zero" power devices, but a NPN (PNP) bjt, with the same base capacitor trick.

A zero base current (or under Vbe) will easily ensure some nA collector current. This kind of devices you can see in car remote controls, where a NPN is connected directly to battery, while applying (OOK) modulation to base.

Use LtSpice to simulate.

See this Low Drain-Source MOSFET Leakage

SIMULATED

Depends on your power levels. From the datasheet you'll get a temperature rise of 62C/W on a minimal copper area, but this goes down to 40C/W with a larger cooling area. A multilayer plane (if you can fit it) with plenty of vias for extra area and copper makes 6cm^2 not all that much. If you look through datasheets for other TOLL-88 packages (or similar, Infineon has weird packages), you'll usually find recommended PCB layouts. At 0.75mOhm rdson, 25C ambient, and Mercury in retrograde you could put 56A through that part before hitting 120C on your junction. The datasheet says you can go as high as 175, but I never take power semis even close to that. (With the minimal copper area you're looking at ~45A to hit 120C).

If this still isn't enough for you, then start looking into heatsinks. The whole point of these packages is to save space, and heatsinks do not do that...

Of course, my analysis only works if you are not switching frequently as it does not take turn-on and turn-off losses into account. It assumes your max ambient is 25C, etc, etc. I'll leave calculating your actual power consumption to you.