Shifters like you envision do not exist.
Also, most normal bullhorns and TT/tri base bars use the same bar diameter for the control areas as drop bars so all the same things can clamp to them, 24.6mm. Trigger style shifters for this clamp diameter also do not exist; they use the 22.2mm flat bar diameter. In many cases you can hog out the radius a bit if you want with files and sanding. Forcing it is a very good way to eventually cause the (usually forged aluminum) clamp area to crack. There have been some bullhorn type bars made that use 22.2, but they're weird because they can't go on normal TT/tri bikes.
What does exist that might be a little closer to what you envision are the SRAM R2C bar end shifters that return to center, so that the hand motion is a little more like a trigger shifter (among other purported benefits). These could be a good answer here if having a SRAM road drivetrain matches the rest of what you need the bike to be.
Note that what your question leaves unclear is what you would intend to do about road/mountain front derailleur incompatibility. If you want your hypothetical inverse trigger shifters to work with a road front derailleur, then that's another stipulation in what you're looking for, because most trigger shifters won't (it would need to be a flatbar road model). If you're building up a bullhorn bike that uses a front derailleur but isn't a straightforward off the shelf road drivetrain, using a friction bar-end for the front has the massive advantage that it will shift anything pretty well, including non-manufacturer-intended setups.
If it's a bullhorn only bike (no aerobar) and the finished product looking clean is a goal, it's pretty hard in my mind to get away from using something like the Jtek Aerobrake (again depending on whether there's a way of making the shifters it works with also adapt to the rest of your drivetrain parts).
