I think there are still several possible points of failure that could cause symptoms you described.
Maybe you have corroded contacts, either in the relay, its socket, or along the path from battery ->starter->relay, or a loose connection. Without much current draw you see battery voltage (actually 13.2 would be a fully charged lead-acid battery at rest if all is 100%).
When the circuit is energized as soon as some current start to flow you lose the voltage/current you need to keep the relay pulled in/engaged.
So, I would start with inspecting and cleaning all the contacts at the starter relay, its socket, etc. You could try also checking the resistance of the handlebar switch (with battery disconnected first) to see if it is making contact, but a high resistance one - this could also cause the relay to not stay engaged.
The fact that the starter will spin with the solenoid activated indicates you've got a working battery, and that another possible problem (a binding starter nose gear/bendix) is not the problem.
I am not 100% sure - I think that the starter relay power is switched from the same terminal on the ignition switch as the dash lights, but if it is on its own circuit, a corroded terminal at the ignition switch could also be a possible point to inspect.