These cars are not known for leaks, and if the compressor clutch engages and stays engaged (even if RPMs increase) that you have sufficient charge to at least make
some pressure. So, if it STILL won't cool, then it is not making pressure, which means: A: the compressor is shot, or B: the expansion valve is stuck 'wide open'. But here is the deal, on these cars, even if the expansion valve was stuck wide open, there should still be some pressure drop across it... so the system will work, just not very well.
This all sounds to me like your compressor is just plain shot. Which is not super common, but it does happen (it is Japanese, after all
). Do NOT get a Chinesium no-name compressor! Only get a genuine VW unit or an aftermarket Denso or Sanden compressor (I am pretty sure the BHWs use a Denso... but I would have to look to be sure... it will be on the tag). I think VW list price is around $550, and the part number is 8D0-260-805-R but do not take that as fact until verified by VIN and the tag.
Keep in mind, this is a best guess without actually seeing the car and placing gauges on it. I'd also replace the receiver/drier, part number 3B0-820-193. Not sure who supplies these on the B5s, but Parker is a possibity, or Hella. I'd only get that from Volkswagen, too.