I have a 1974 FLH that's been giving me problems for a few months now. I get a loud click from the solenoid but the starter does not engage.
Here's what I've done so far; inspected and replaced solenoid (twice), rebuilt starter, replaced starter relay, ensured the battery is fully charged and checked out all wiring related to the starter and its components. All connections have been checked for proper resistance, cleaned and appropriately tightened yet I still get nothing but a click.
Any assistance would be appreciated...

 

When you put your volt meter on the starter solenoid terminals, what are you getting?

 

Just take a Screwdriver and stick it between the long post and the small [engagement] stud. Don't use yer favorite screwdriver for this...use the one in yer Bitter-half's tool drawer. If it don't turn over properly...yer Battery sucks. Assuming you have good connections and proper cabling.
Or...there may be something physically wrong with yer set-up...like forgot the thrust washer when assembling the outer primary?
If it worked before, and you have Not changed anything in the Primary...
Well...return to the above.

 

You need to load test the battery, until you are sure that the battery is not the problem, you can`t properly chase down the problem.

 

I'm getting the same reading I would get from the battery itself. In this case 13.65
I'm taking the voltage from the big terminals with the ignition turned on.

 

I actually have an old feeler gauge rigged up to do the same thing, but still get nothing but a loud click. I can feel the solenoid engaging, but not the starter.

 

The initial reading on the battery was 14 volts when I took it off the charger yesterday. It reads around 13.65 without load and drops to 12.30 when I turn on the ignition and hit the starter switch.

 

If it is holding 12 anything with the Starter engaged...it is a damn good Battery.
Been into the Primary recently?
Did it just suddenly quit working out of the Blue?
Jump 12v to the starter motor without engaging the Solenoid. Does motor spin strong?

 

If I read everything correctly, the starter doesn't spin, so the voltage wouldn't drop.

To trouble shoot these, you need to understand how it all works. The solenoid serves 2 purposes. It pulls the shift arm in to push the bendix gear into the flywheel, and it closes a set of internal contacts to allow high current to pass from the battery cable to the starter motor. With the bike off, you should get 12volts at the log stud, as that should be directly connected to the battery. When you push the start button, you should see 12 volts at the small stud, the solenoid will shift, and then the smaller large stud (which should be wired directly to the starter) will see 12 volts. If you are getting 12 volts at the shorter large stud, it's a starter motor issue. If you don't see 12v at the small stud, the issue is in the relay (or start button). If you don't see 12v at the short large stud, it's the internal contacter in the starter solenoid.

 

If it is holding over 12 volts when the starter button is pressed, I don`t think the starter is even drawing current...

Take the battery to a shop and have it load tested. Right now you are trying to load test a battery with a suspect starter circuit, you are just chasing your tail.

One step at a time, don`t assume anything is good until checked, or you can just jump from one thing to another and drive yourself nuts.