The coach is likely to be your best bet for this one. Of all the people I know who live or have lived in this city, all have found it cheaper to travel to Heathrow, and in most cases London itself, by coach. The 403 coach runs roughly twice every three hours from Bays 1 and 2 at the bus station, and usually has some fairly good deals going for Heathrow. If going by train, those are tough to beat owing to either (a) the ridiculous access charges imposed south of Airport Junction or (b) the requirement to travel by coach anyway from Reading.Are you prepared to be tied to specific trains? Are you prepared to travel part of your journey by coach?
The Piccadilly line seems to go via the land where time stands still when you're going to Heathrow. Piccadilly Circus to Heathrow T5 takes 53 minutes, apparently, but it feels like much longer.What about changing in London and getting the Piccadilly line over to Heathrow T5?
The problem I came up with was that the return time is not Off Peak.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.