The lamps might not care but the 16 MOSFETs switching the current TO the lamps do "care" in the sense that if you get the polarity wrong their body diode will conduct and all the lamps will turn on. That might serve as an indication to the user that he just did something stupid...
If that LCD has a backlight it might pull some current.
PCA9865, what's that? Tried to Google for it but all I seem to get is 9685 which is a 16 channel LED dimmer IC, is that what you're using and if so are you driving the MOSFET gates with its output?
The P-channel MOSFETs I suggested are just two I picked, like I said there are hundreds, if not thousands, to choose from. As for the fuses you may want to look at using a PTC.
/Henrik.
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