Are you prepared to be tied to specific trains? Are you prepared to travel part of your journey by coach?
What about changing in London and getting the Piccadilly line over to Heathrow T5?
Could I suggest a slightly simpler route : as above but to West Drayton. Then bus 350 to T5 departs from outside the station ( timetable : http://www.londonbusroutes.net/times/350.htm ) about every 12 - 20 minutes depending on time of day. Journey time 15 minutes door to door.
BTW its £41.85 return YP railcard Bath to Heathrow T5 (HWX) via Reading and the direct RailAir airport coach on dates and times specified. That's one change and coach to and from terminal door. If you have bags that's a lot less hassle.