Sorry, I'm busy at work a lot of the time at the moment, and in a totally different timezone to the UK - so trying to catch up as/when I can...
The ignition switch doesn't switch a live feed, but rather it switches a ground signal. So on a multimeter, you possibly will see a voltage on the wire when the ignition isn't switched on, and it should go to 0V when turned on if it's connecting to ground properly.
That being said, depending on the resistance of the connection will depend on whether the ground signal gets to the BECM properly or not.
I would still be trying a CONTINUITY test ALONG the white/pink wire from the ignition to the BECM. Unplug the connector from the BECM with the White/Pink wire in it, and do the same at the ignition. If you have a digital multimeter with a continuity test setting (or beep tone - same thing, or if you don't have that function, then resistance check, on a 200 ohms setting) then (probably with a small bit of wire as a jumper to get into the pins) connect one probe to the ignition end, one to the BECM connector end. If it beeps and shows 0 ohms, then you have continuity and a decent connection. If you then wiggle the wiring, wiring loom in the sill etc and the meter stops beeping, then you have a break in the wire. If it stays beeping and the resistance is still showing near 0 ohms then chances are the wire is OK and the fault is elsewhere.
If you do that, then it will tell you whether the issue is in the White/Pink wire from the ignition back to the BECM. If the problem isn't there, then remove the meter probes, plug it back in, and then turn the key to ignition position I whilst listening down near the BECM to see if you hear a click of the relays when you turn the ignition on.
If the wire is working, then it does point to an issue in the BECM, or something that's shorting out somewhere and causing the chattering of relays that you hear when the problem occurs.