Hey!
you could try some of these routings...
BUD-FRA-SIN (last part of QF 744)
BUD-LHR-SIN (QF 380)
For Cathay seems you could route via AMS, FRA, LHR, FCO (I think is a two cabin plane)
JAL: FRA, CDG, LHR all seem to have valid connections...
I would try to get the 380 personally and then try CX First (or Viceversa)