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If you want to get it going you are going to have to add a Starter Relay and mount it securely (I recommend the Bosch type) . Find the wire from the handlebar start button and wire to this diagram.
I think it a rather stupid design to make the relays in that assembly nonremovable.
Battery is brand new and has 350 CCA, it’s not the battery… what can cause a “click”? Is there a relay somewhere that controls the starter?
If you're hearing the click the relay is working. Just because the battery is "new" doesn't mean it's good. Put it on a charger, not a tender," for a couple of hours then do a load test. If the battery checks out good it might be time to replace the contacts in the solenoid.
I chased the same problem for over three years on my ;93 Heritage. So over time, I changed the starter, changed the ignition switch twice, changed the battery three times, replaced the solenoid contacts, replaced the starter switch and cleaned all the plugs in the wiring harnesses added a push button cover on the solenoid and still had the problem. When the new starter was about two years it too went dead so
I installed another starter and finally pulled off the tanks and eventually found corrosion on a wire in the starter circuit. No more problem.
