I'd first check AC power to the MPS-50. Make sure it is good.
If the AC power into the MPS-50 is good, just replace it. They are ~$80 "bricks" that are outside of the unit. There is no install. It just gets connected to the AC and to the DC and then plugs into the back of the unit.
If the refridge works with either AC or DC it is working. It knows nothing about the source of the power coming into it. It expects good power. It, itself, doesn't do any conversion or much conditioning.
You may be seeing good voltage and still having a problem because the waveform isn't right. You'd need an oscilloscope to see that.
What I mean by that is that the brick should transform the 120v AC into a lower voltage AC, then rectify the alternating current into an all positive bouncing waveform and then smooth that into DC. It is possible, for example, that the rectifier is half bad and only the positive or inverted negative wave is being produced. It I'd also possible that after being rectified the smoothing part of the circuit is bad, e.g. open capacitors. Your meter would likely still show the right voltage in either case -- but the compressor wouldn't run. You'd need to be able to see the waveform to know.
I haven't seen the schematic for the MPS-50. There are other possible designs. But, if the refridge runs on DC and isn't running on AC, thst bricks is overwhelmingly most likely to blame. Making the AC and DC look functiomally the same, like the correct output for the refrigerator, is exactly the purpose of that brick.
Could it be something else? Sure. The controller in the compressor xoukd be hyper sensitive to something thst shouldn't matter and happens to vary depending upon the input source. Etc. But these scenarios are unlikely.
The brick is ~$80. Order it from Amazon or West Marine and, if I'm wrong, the return will be easy.