I'm having the same problem with my 06 FLSTI. I was wondering if the gel battery was the fix cuz I did the same thing by replacing the battery first and my new battery is also an acid one. It started up just fine the first two or three times. However, this morning I rode in to work with my Aux head lamps on and when I stopped to fill up it did the click sound again. I just turned off my Aux lamps and it started up.

 

I still think it's the battery. BNut, if you want to pull the starter here is how to do it without removing the oil tank. Gtet a long extension and a socket allen. The front starter bolt is cake. For the rear you need to move the oil tank a tad bit and make sure your extension is a wobbly extension. You can get the allen in the bolt head and get it out. Leave the bolt in the hole and replace the starter and put it back together. I avhe done it this way a few times and it was much simpler than pulling the oil tank which is a PITA. I avhe pics of this somewhere I'll have to dig around and see if I can find them.

 

+1. Being able to be jump started says everything is working except getting the volts from the battery to the starter. You say it read 12.4 volts? That sounds pretty low. It should read like 15 or so at rest. My gut tells me either bad battery or your charging system is not charging properly.