Eureka!
DPD (Dead Peer Detection) is the mortal enemy of Keep Alive, using both at once is a problem of sizeable scale. Since you don't want your tunnel going down ever remove DPD.
You would see it in Status->Routes, but only while the tunnel is up. A better place to check would be if you have both that check box, and the automatically add route box on the advanced tab checked.
As this is a L2L installation you should have the one on the advanced tab only and not the one on the routing tab.