I used to own a 1996 323i coupe which had the same problem. If you really do want to roll your sleeves up then you can find the offending sensor using your multimeter resistance measurement.
Its been a while since I did it but you have to unplug the ABS sensor on each wheel which means you'll probably have to take the wheel off first. On mine, there were 3 pins on each sensor but one of the pins didn't return a value (screening connection?). The other two pins measured (from memory) about 1 Kohm but the faulty sensor measured something like 1 Mohm. Sorry I can't remember which pins so you'll have to experiment.
Once you've found the offending sensor(s) then you'll have to remove it which usually proves difficult as the retaining bolt is probably rusted up and seized.
Once you have 4 working wheel sensors again, the system detects this and extinguishes the ABS and traction control lights after you have had a drive.
I hope this helps and good luck.
I had heard it was possible to use a meter and now that you have said it too, I'm gonna have a go. I work with electronics for a livin so I'm quite handy with a multimeter. Cheers bud, I'll have a crack at it. As for gettin dirty, I don't mind, it's a hobby of mine lol