I"m sure you've searched other threads.
Don't JUMP start your DOG!
I have the WP, and the RUN ON/OFF button does cause some issues if you depress it. Even with a key reset, it sometimes won't let me reset my bike to start up, I usually hit it by accident coming off another bike. But mine doesn't CLICK, it just does nothing.
Anyhow, I believe a blinking green light is not normal.
Check the CCA's off your battery FIRST. Don't assume because it's a NEW battery it is putting out enough juice. These batteries sit on shelves in warehouses...
Also, check the big STARTER GROUND. Boomer4444 had a similar issue and did all kinds of tests and found that his starter ground was securely bolted on, but the bolt was tapped out on the inside and didn't make a good connection. Check all your grounds and make sure you have good connections.
It really sounds like it's a ground/power issue to me. I had issues with mine, and downloaded the WP374 wiring diagram, and the BDM diagram and traced EVERY wire down, did continuity tests etc...to fix all my issues.
http://www.bigdogbiker.com/forums/technical/48064-wire-plus-install-2004-chopper.html
http://www.bigdogbiker.com/forums/help-wanted/30485-what-do-ehc-lights-mean.html
Edit: Does it click when you hit the START BUTTOM or before then? Mine does the following. I turn the key on, the headlight is on high beam, it clicks and goes to low beam. My light on my box BEFORE I hit START, after turning on the key is SOLID RED. I'd fire it up to see if it changes, BUT it's -10 right now....hahahahaha