First, check the power supply. Although it may be specified to deliver the required power, it is possible that the power supply has failed in a way that it can not deliver the rated power. At lower load, the voltage may be correct, but under higher load, it either droops or cuts out completely.
To check this, use a voltmeter on the power as it enters the CPU board, not where it leaves the power supply. This accomplishes one additional check. If the voltage droops rather than cuts off, it may be that the connections have corroded and have a higher resistance.
If you have any kind of oscilloscope, I would recommend it over a simple voltmeter, because the power interruption or droop time may be very short. When the CPU resets it will switch off the load that causes the problem, and the power may quickly resume the correct value.
Second, check that there is not a short in the bed wiring. You might detect that with an ohm meter. You have used two different CPU boards, to it is unlikely to be a common fault on both boards, but you might be using the same wiring.
Third, check the routing of the bed heater wires to see that they are not near other wires which connect with the CPU, including thermistor wires and wires to the UI. High-current switching in the bed wires could be coupling into other wires and conducting a RESET signal to the CPU. Ideally, the heater wires will be twisted together with about 3 (or more) twists per inch, and not twisted together with other wires.