Next: I'd decided to swap out the stock white 8-hole pickguard with a black 5-hole one:
For convenience, I still want to have the pickup mounted to the pickguard as per Simon's earlier suggestion, so removed it from the white one and put it in place:
Used the pin vice to drill again, working my way up from 0.5mm to 2mm bits; first pilot hole from the back so that I could drill it through the pickup's mounting hole, then the others from the front so that I didn't get ugly visible tearout. First hole done and screw in place to test:
And second done:
Much easier than I'd feared it would be.