Stratis

The issue here is with the nature of your 'measurement' waveform; a soundwave is distinct from a light beam in a very specific manner; it is not at the greatest possible speed within its medium.

The medium of sound is whatever physical medium is carrying it; if it is air, then it is Mach 1. This however does not equal 'c' in your equations, as it is possible to exceed the speed of sound in any given medium. You falsely attribute the character 'c' to mean Mach 1, when one of the governing principles of the letter c in physics is that it is the upper asymptotic limit for all velocity within that medium (for light, a vacuum), not just the wavefront itself.

Also, if you are using only a single reference frame then STR basically ceases to apply; we here on Earth are hurtling through the cosmos at quite a velocity yet notice no dilation or synchronisation issues here because we are all in roughly the same inertial frame. STR only applies when two bodies are in relative motion. If your second observer is on the platform then the synchronisation and dilation issue returns with a vengeance.