I opened the A/C and located the compressor relay on its main board as shown below.

The solution was to change this normal relay with a solid state relay. Because of the way a solid state is constructed, the output signal always starts at a zero crossing. This means zero volts going through the compressor coil and therefore zero initial current. I highly recommend checking out this video from GreatScott where he explains this in more detail.

This is a screenshot of the compressor voltage with normal relay.

And here with the solid-state relay.

As you can see it clearly starts at zero crossing and that solved the problem!