Electronic – Low Voltage DC Automatic Transfer Switch

The problem with your +3V source is that it is ground referenced.

A MOSFET turns on based on a voltage differential between its Gate and Source pins. If you have access to a 15V supply, that'd turn on the FET if its datasheet calls for 3V Vgs.

You can use a schottky diode to prevent a voltage drop in the +5V rail.

If you're using a physical switch on the 3V, why don't you just use that switch on the 12V with a schottky?

You'll notice that in Figure 2 it states "Ic/Ib = 10".

This means that to saturate the transistor you should drive it as if the Hfe is only 10, not the almost 100 in Figure 1. Note also that figure 1 is the typical figure not the guaranteed value. The guaranteed value in the data sheet is only 10 at 3A collector current.

You need to provide more base drive.

The PIC however will not be able to drive the required 180mA output. You need a darlington transistor such as a TIP120 or these days a better solution is to use a MOSFET. The TIP120 would have a saturated voltage drop of ~1V and so would require a heat sink. A mosfet would have a very low drop and would work without one.

Even there you have to be careful as many MOSFETs will require 10V drive, more than the PIC can provide. For this there are so called Logic Level FETs that switch with only 5V on the gate.

Something like FQP30N06L would work.