We've flown TPE-LHR return the last two winters and have visitors coming this winter. Unfortunately the prices are just what they are, we also had to fork out a decent sum both times.

​​​​​​Is TPE and then taking the HSR or Mandarin/Uni airlines an option? I know if you fly long haul BR and then have a domestic the next day the overweight luggage fee is cancelled (we use this as we fly to an outlying island). Obviously you'd have to get to Songshan but overall cost might be cheaper.