........ From the way it acts I think there is a loose connection somewhere, but there are so many connections on it I don't know where to start.
From the sounds of it, replacing the battery was a necessary first move! :thumbup: But I reckon that in doing that, you've done what I did when I fairly recently replaced my 2013 RT's OE battery...

I moved the Negative cable around a bit while getting the new battery in, and that was enough to loosen the other end of that cable. In my case, that resulted in a 'not really great' & intermittent connection on the 'non-battery' end of that cable which meantan intermittent & alarming DPS failure! :banghead:
Admittedly, your 2012 probably has the battery under the seat rather than behind the frunk like mine, so yours might not be exactly the same problem, but it's a good place to start! First, check to make doubly sure that the battery terminal connections are tight & making good contact - nb: I've seen 'tight' terminal bolts that simply didn't clamp the cable onto the battery terminal at all!! :shocked: Those battery terminals with bolts that go into a 'blind hole' can be a real trap; bottom out the bolt in its blind hole & it 'appears' tight, but the ferrule on the end of the cable can still be loose & flopping around in the breeze with that 'tight' bolt going thru it - so make sure the
cable end IS actually tightly held onto the terminal with a clean & firm connection - star washers work really well at this!

hyea: And then follow both the + & - cables away from the battety, checking that their connections are tight & clean on their other ends! While you're doing that, check &/or clean every other 'electrical ground bolted/screwed to frame' connection that you can see/find & any other 'bolt &/or nut' type electrical connection you can see; those in particular can & do develop a little corrosion between the metal surfaces as time passes, and that means increasingly poor electrical connections - adding star washers to all those connections too can resolve any problems now and help save a lot of grief down track! :thumbup:
Good Luck! :cheers:
Ps: IIRC there's a 'wiring earth to frame' onnection under the seat on the 2012's & earlier that has a 'pressed nut' pushed into the frame cross piece, and if you undo that bolt &/or over-tighten it, you can push or pull the nut out of its seat & compromise that ground conection! It's a while since I played with one of these, so once again, it's down to memory...

.... still, IIRC it's a real bear to fix once you've loosened this connection, cos you've somehow got to remove the bolt with its now freely spinning and largely inaccessible hidden nut before you can re-fix it so that ground connection stands any chance of every working properly again.... :cus: I think many resorted to using a large self-tapping screw directly into the frame cross rail (in a spot cleaned of paint too!) instead of trying to replace the bolt & its now permanently loose nut that formerly held the ground connection tight?! :dontknow: