I am starting to wonder if the six caps on the Amazon sourced relay board are not to blame. Below is a probe when the battery is disconnected, bench power supply on IN1, DC load off. I switch the bench power supply on and record the voltages. Yellow is probe at OUT, cyan is probe at IN1.
It seems like the rise on IN1 (cyan) corresponds to the 6 onboard capacitors charging up. When they exceed the voltage threshold of the relay coil, the relay turns on. This connects IN1 to OUT, and since OUT has no load the capacitors are not discharging through the load, however they have discharged through the coil of the relay to store energy in the magnetic field. However once that was done, the caps charge up again and stay charged, no drain on the input.
When I place a 35W load on OUT, and then turn on IN1, I get below:
Here the same initial charge cycle occurs for the onboard caps. However once the relay closes and places IN1 on OUT, the caps are immediately drained. That coupled with the drain from energizing the coil causes the input voltage to be too low to keep the relay coil energized - thus it opens. Once it opens the caps charge up again, closing the relay etc.
Does that seem like a reasonable explanation?
** UPDATE: I desoldered the caps - this helped, I can now place 10W of load without it oscillating, but 15W+ is still too much. So there goes that theory.