I haven't seen any indication that it's going to cause any sort of problem. The 1301's trigger has the sear on it:
...sorta kinda in the same ballpark as the AR15's trigger group. I don't know of any long term issues with the AR being stored in a cocked state, and in the time the 1301 has been out I don't know of anyone who has encountered a problem resulting from having the gun stored with the hammer cocked.
The reason I store an 870 with the hammer down on a closed bolt is because I don't want to have to fool with the action lock to get the gun into action. A lot of police departments mandate that the action be locked as a part of their setup procedure and it's not uncommon to see officers struggling to get the gun into action. I'm trying to avoid that for clients.
My teaching to them...and it's documented on video...is to ensure there are absolutely no shells in the gun
*period* when they press the trigger without intending to fire a shot.
My advice on the 1301 would be exactly the same. I do not want to press the trigger on a 1301 that has live ammunition in it unless I intend to send a payload into something...because sooner or later it will get messed up and that's exactly what will happen.
It feels wrong to do it because you're pressing the trigger on a loaded gun and hoping it doesn't fire. That feeling of discomfort is your brain telling you this is a bad idea. So don't do it. Even if leaving the hammer cocked killed the hammer spring...which we have absolutely no indication is the case...I'd much rather replace one every now and then than set myself up to send a load of buckshot somewhere in the house.