Possible start capacitor, or maybe the compressor?

I have a 2005 model Kenmore refrigerator, model WPW10662129. The unit recently has started either a capacitor or a compressor click that happens at random. The scenario:

The fridge is running fine, both top and bottom compartments are at temperature, then a loud, heavy CLICK can be heard from the compressor area. The fridge skips a quick beat, then continues on. A few minutes later, the fridge shuts off. 30-45 seconds later its back on again. This time I can hear the fan running in the freezer compartment, but not the compressor. A little later, there’s another heavy CLICK, and I can tell the compressor as started. 5-10 minutes go by, the system shuts off, and the cycle repeats. Sometimes there sounds like a the fridge makes the same CLICK sound even though the compressor is already running.

The fridge is never off completely for more than 30 seconds at a time.

Could this be a startup capacitor problem, or is there a possible problem with the compressor?

I’ve researched the replacement of a ‘hard start kit’ but am unsure exactly how to make sure I have the right type/kit.