If an A320 loses an engine why does the landing category go to CAT3 single?
According to the forums PPRUNE, for A320, taken from here:


  The power supply split function that monitors the electrical segregation of the FMGC's does not consider the APU generator as an independent electrical source. As a consequence, when an engine generator is lost the capability is downgraded to CAT 3 Single.
  
  The electrical power supply split function monitors......
  -The two BUS TIE contactors
  -The DC TIE contactor (between DC BUS 2 and DC BAT BUS)
  
  When one of the above contactors is closed, the FMGC considers that the power supply split is lost (independently of the APU generator)
  
  The A330 does not have this restriction and can be CAT 3 Dual capable even with one engine generator failure as APU is considered an independent electrical source.


Also- if you dig in the FCOMof the plane, you would find that if you perform dual engine cat3 approach, and an engine fails- the buses remain split to prevent the failures from reaching the other side, and it degrades to SINGLE.

If you perform an approcah with the APU online- the electrical system is built that if a failure occurs- the bus ties close and power is managed to the whole aircraft from 1 source only- meaning the buses are not split. Hence the single.