If one were trying to optimize for minimum cost then there's some trade to be made between mechanical complexity and the number of sensors. Given that the LIDAR sensor is comparatively expensive vs ultrasonic sensors and (arguably) servos, there's some sense in going with a mechanical approach. On the other hand a mechanical system is likely to be more fragile, particularly in a hobbyist one-off build. Also the paper you link describes a system for monitoring the rise of multiple individual loaves in the commercial bakery scenario, so a considerably more complex scenario than the current problem.
With regard to the range accuracy in the original post and without have played with the numbers, the 1-2 mm accuracy seems overly tight. That said, I probably take my bread making less seriously than you. Also the system you've described works for the first rise. Are you planning a monitoring system for the second (presuming that applies)?
As an aside, I was surprised that one consequence of the pandemic response is that bread flour and yeast seem to be in short supply at the grocers in my region (central New York state in the US).
Well, it looks like I can get get 10 off VL6180X for about 26usd from Aliexpress, so not worth having a complex mechanical system.
Initial dough vol about 1600ml. 1mm level equates to 50ml, so +/- 1mm gives +/- 3% accuracy, and 2mm, 6% - that's why I picked those figures; aim high, accept a bit lower if necessary.
Final proof (loaves) are not such a problem for me, as I usually retard overnight in the fridge; for room temp final proof, the finger poke test is reasonably successful.
And yes, all flour is scarce as hobby horses' poo at the moment - we are now a world of home bakers! Which is nice as long as all that flour isn't sat at the back of the cupboard for evermore.