What local pressure changes would a starship launch on Mars cause?
Here's my non-expert on this.
A local pressure disturbance in an atmosphere will move or relax at pretty much the speed of sound. It's coming out of the rocket at 3000 m/s from say a 10 m cluster of engines but will quickly randomize and expand at 300 m/s, which means at a distance of 20 or 30 meters it will be able to deplete at the same rate it's being fed with new gas by the engines. So I think this will certainly be measurable and characterized as an infrasound event and infrasound detectors elsewhere on the planet would "hear" it.
For some background on the topic of sounds on Mars and low pressure environments (e.g. in space) refer to:
Infrasound detectors are basically high frequency response barometers but then again perhaps so are some kinds of microphones (arguably?).
Per Infrasound; Detection and measurement:
NASA Langley has designed and developed an infrasonic detection system that can be used to make useful infrasound measurements at a location where it was not possible previously. The system comprises an electret condenser microphone PCB Model 377M06, having a 3-inch membrane diameter, and a small, compact windscreen. Electret-based technology offers the lowest possible background noise, because Johnson noise generated in the supporting electronics (preamplifier) is minimized.
Not all infrasound detectors are made from microphones, barometry can be done many ways.
The circular pattern of inputs can be sampled with separate barometers and the timing between pulses can provide direction information.
No, not on Mars (yet!)
Infrasound arrays at infrasound station IS18, Qaanaaq, Greenland. Part of the Nuclear-Test-Ban Treaty Organization Preparatory Commission monitoring system