Drop bars and trekking bars move your hand positions in opposite directions relative to the headset. So a drop bar bike generally has a shorter TT to account for the longer reach to the hoods/drops. A trekking bar bike generally has a longer TT to account for the shorter reach to the brake/shifter portion of the bars. And a flat bar bike would have a TT somewhere in between.
I would mock up the rough position of the controls on the trekking bars you are considering and see if the TT on your BF works with that position before you invest in the needed parts. The Bike Friday H-bars offer some of the benefits of Trekking bars and would work with similar TT lengths as drops.