With STI's not sure. Back in the day there were bar top brake levers for while climbing or something I thought, a dual set. But you could mount normal base bar brake levers in the ends of the extensions instead and run dual levers. Likely would be a mess to setup.
IMO this is just one of those inconveniences of clip ons as a really good budget item but not the 100% best thing. Not worth the complexity IMO to do it.
You have to think, since you can't shift from the clip ons braking doesn't gain you much as you'd have to get out anyway to shift from having braked. The remote brakes would really just be an "oh no!" emergency kind of thing versus useful.
If you own a Di2 or other electronic shifting road bike though, folks do mount remote shifters to the clip on extensions............making it a really slick setup.