That's the side effect - Piercross/K&N etc filters DO rely on the oil to catch the dirt. Sadly the oil will trash a hot wire AFM - maybe not immediately, but its lifespan will be shortened by a horrific amount...
Think about it - the dust particles need to be retained somehow. Paper filters are disposable for a reason. Foam/cotton ones use oil as the air holes are bigger (we are talking microns here, BTW) and are oiled as the path through them for the air is convoluted. Dust doesn't like travelling round bends (unlike air), so gets stuck in the oil.
Oil does get transferred off the filter and into the airstream. As all air goes through the AFM, some gets stuck to the hot wire air flow sensor. Note the word hot...what happens when oil hits a hot metal surface? Look at a proper chinese wok.. :-?
When the hot wire gets coated with oily residue (the same muck you have to clean off your throttle body occasionally), it reduces the cooling effect of the air that passes past it - air that the ECU tthinks is entering the engine - so the ECU injects less fuel. Good? Until the ECu thinks there cannot be this little air going in, so flags up a MAF/AFM fault or two...or a Lambda fault, as the weak mixture measured at the exhaust can't be corrected within its limits, so you get fuel trim faults..etc, etc, etc...
When I need a higher flowing air filter on a car, i looks at a car with something bigger and use that cars larger filter. Not a Mig, but Mini magazine did a test with the stock filter (and housing) vs every other filter on the market. The standard one won by a comfortable margin..
OK, you can't replicate the noise easily, but the old airbox drilling technique still gives the noise as long as it can still get cold air!