This suggests you can do it by post or phone.
Alternative is go to a tube station (which is a bit of a sod if you live in SE London and are some distance off the end of the underground...) although of course this option won't be around for long if Boris has his way.
If you want to do it in person, yes, the shortest journey would probably be SWT to Feltham then bus to one of the Heathrow stations or Hounslow, or change to a Hounslow loop train then walk / bus to Hounslow Central (looks like it's about half a mile from Hounslow SR Station.
Or Richmond then one stop round to Kew Gardens (I'm assuming that Richmond, although served by Underground and Overground trains, has a SWT ticket office.)
Note that it says that to do this at a tube station, you must take the debit / credit card you originally added the credit with. Not quite sure what happens if you've changed credit / debit card since then.
Unless I was travelling close to an underground station in the first place, I'd be very inclined to try and do it by phone / post - also avoids the risk of the ticket office you end up at being closed due to staff shortage or technical problems!