Electronic – Switch (dis)connects an LDO regulator from VBATT. Where should the filter cap go

You have 500 mV headroom, so a LDO should be possible. A diode is not a exact voltage drop. The output of a LDO will be much better regulated. Unless this is a very high volume product and the extra few cents actually matters, use a LDO.

You just cannot add gain to the output of an op-amp (extra transistors) and expect this to not oscillate without the appropriate compensation.

Look also at the data sheet of the NJM4559 op-amp that was giving you problems. There is nothing in that data sheet that even hints at what the phase margin is. I would expect to see a graph like this: -

This is for a TL084 op-amp and you need to study it in detail; at low frequencies the phase difference between output and input is about 180 degrees - this is what you'd expect from an amplifier i.e. it is largely inverting. As frequency rises there is usually a mini plateau were the phase angle has shifted about 90 degrees i.e. it is behaving like an integrator.

As frequency rises further, gain drops to unity and the phase margin is about 50 degrees i.e. it's 50 degrees away from being an oscillator.

Some op-amps are a bit tighter than this and for many op-amps, the unity gain point (the highest frequency that oscillation could occur) is a lot higher. For the NJM4559 this is about 6MHz. For the TL084 it's only 3MHz.

So, wiring an op-amp with regular resistive feedback and no transistors is fine for the TL084 and maybe it is for the NJM4559 but, the data sheet doesn't give any indication that it is. I would be very mistrusting of this device given the spec that I read.

Now, adding gain into the feedback loop (the two transistors) is going to cause problems nearly every single time - you are basically shifting the unity gain point of the op-amp up (maybe 20 dB or more) and this might very well move the actual unity gain point on the graph by up to 10x in frequency. You'll probably also be degrading the phase characteristics and now you have an oscillator because the phase margin has massively fallen thru 0 degrees at unity gain.

So, then re-assess what the gain is at 0 degrees and you'll find out that it's several dB above unity = oscillator.

Lower the gain of the two transistors is a good start - put a source resistor into M1 of maybe 4k7 and reduce R2 to 4k7 - this is a start but by no means might this be the only thing you need to do.