What is the current requirements for the solenoid? Can you measure this with a meter in series on current setting connected across the battery if there is no data on it?
Is your supply a real battery or a power supply? A real battery will be able to handle the voltage surges of the coil.
Do you have a scope that you could use on the power supply at the cobra. Seems strange though as the Cobra is 3.3V so a drop to even 5V will not be an issue for the Cobra (and that would be a big drop for a small solenoid)
You say this happens when you energise the coil?
To be honest, a scope on the power line is your only way to test this. I had a similar issue the other day with a bench supply and an embedded PC. As the PC booted it would get so far and switch off. I never saw any spikes on the supplies current meters but a scope on the input to the board showed a drop to 3.9V and this was causing the shutdown. Using heavier supply cables sorted this but the scope was the only tool I had that would spot this. A valuable asset to any electronics workbench.