Thinking about the pulsing voltage, two things come to mind. The compressor is not getting enough current so the compressor stalls and kicks out, first taking current and then dropping out causing the voltage pulse. The second, which I think more likely, is the battery charger is regulating the voltage to the battery by varying the length of the pulse. On some battery chargers, you will see a yellow LED (light emitting diode) flick on then to another green LED back and forth. This is a common way for an intelligent charger to regulate the voltage. Although somewhat controversial, the idea is the pulsing type of intelligent charger can reduce sulfonation better than a constant voltage type of intelligent charger. Even if there is no ground fault or resistance in the wiring, the current from the charger is sufficient to cause the voltage pulse. Disconnect the shore power and see if the pulsing still exists. There may not be a problem in this case. If pulse still exists, consider compressor stalling.
However, in your situation, the voltage drops to 11 volts, which is lower than what the house battery would be capable of supplying, it should be above 12 volts. Take the volt meter and put it on a ten volt range or maybe 12 or 15 depending on the meter, place one probe on the house battery positive post, the larger post, and the other at the wire going to the frig. If you see a voltage reading of more than half a volt, you have a high resistance somewhere between the battery and the wire to the frig. Next thing is to narrow down the problem with a test from the battery to the post on the positive side of the circuit breaker, or the fuse, then across the fuse or circuit breaker. Surprising, many times there is a high resistance in these circuit protection devices so replace as needed, being sure to check for corrosion where the fuse of circuit breaker plugs into the power distribution panel. You should also test the ground doing the same thing by holding one probe on the negative battery post (smaller post) and the other end to the ground at the panel with the fuses or circuit breakers (the DC panel). It might be that the wiring is undersized. Some designs have as much as a 10% voltage allowance. Others are about 3% voltage drop. If you have no voltage drop across the terminals to wire connections, consider undersize wiring if you get a reading above half a volt. Sometimes the terminal is not crimped correctly to the wire and you get a significant voltage drop.
The reason you are seeing a constant voltage of 12 volts on some of the other grounds might be that you are measuring the voltage from the starting battery. Most marine battery chargers will have more than one output, one for the engine battery for starting and the other for the house battery, which is everything else. If the battery charger is setup correctly, you should be seeing about 13.4 volts on the other ground where the engine battery might be fully charged and therefore you see no pulse, otherwise you would also see the charging pulse. Most likely your engine battery is not hooked up the charger and has deteriorated from sulfation. Buy a hydrometer like this
If you see little bubbles when you draw fluid into it, tap it a few times to get rid of the bubbles as it can influence the reading. If you buy the little cheap plastic hydrometer, the bubbles will not come loose from the plastic with tapping. The bubbles will come loose from glass. If the battery is more than half discharged, buy a new one as it is not possible to get a full charge into the battery, unless the discharge is recent, say within several months. If you are using an automotive battery charger, the boat is likely not isolated from the AC shore power and you could get electrolysis, a severe corrosion problem. In this case you could consider an isolation transformer which would allow AC power in the boat for other purposes besides an automotive battery charger