I imagine one could build something with thin capacitive sensors under the bar tape (or the rubber of the hoods) and a microcontroller. Here is a board with 5 touch inputs you could simply wire up if you have some basic knowledge of electronics and microcontroller programming.
Alternatively you could record the ride on video (mount a wide-angle camera to the bike somewhere) and manually analyze it (or train some machine learning model to distinguish between the three main positions).
Maybe even data from an orientation sensor or accelerometer on the back of the rider would be sufficient to distinguish between the positions. Try putting a smartphone into your jersey’s back pockets and record the sensor data.
However, I doubt the end results would be very useful. Which hand position you pick highly depends on terrain, wind, intensity of the ride, geometry of the bike (and drop bars), the rider, your seating position etc.
I think generally one mainly adjusts the seating position for the hoods because the assumption is that you’d spend most of the time there.
Also keep in mind that even though there are only 3 main hand positions there are countless variations. For example there is a big difference in gripping the drops right at the end (feels like your shoulders are almost above the hands) vs. at the bend of the bar with elbows bent and a very aggressive riding position.