I've done countless double to triple swaps with both DT and stem friction shifters and never had to change the shifter. I've also gone from a 5 to 8 speed on the rear without changing the friction shifters. I'm not sure how much travel the 6 speed Index shifter has in friction mode, mine are all packed away, but it should at least be enough to handle an extra cog.

3 or 4 years ago Shimano 3 X 7 indexed DT shifters were cheap and plentiful on Ebay, I see that is no longer the case.