Hahahaha...This thread just keeps getting better.
With how the OP described the problem I would never have been able to diagnose it.
A reversing valve needs a differential pressure to shift. If you can't make this happen it won't shift.
Glad you figured it out.
Also, hope it's not low on charge when it's time for cooling.
When you say the compressor starts for a second then shuts down, I am assuming the contactor is not opening and the compressor is going out on thermal. This combined with the info that you are checking the charge in cooling mode and the only value we know is 8 degrees subcooling makes me wonder if the system may not be charged correctly.
You stated there is no TXV so when you are checking pressures in cooling mode we need to know superheat, however, the more accurate method to check the charge would be to look at the manufacturer winter charging chart which should be with the unit unless the installer didn't leave it there.
I have seen reversing valves shift with no noise, and, keep in mind refrigerant flow is what causes it to shift, so the unit must be running for the shift to take place.
If the compressor is out on thermal it may take several hours to cool enough to start