OK, I compared the parts diagrams between the 2012 EX250J and the 2013 EX300 and made the following observations:
The front fender may swap, but it wouldn't be a flawless swap. Looking at the metal stay fender mount, (35030-0006 on J-series, 35030-0055 on 300), it appears largely the same except for a few obvious differences. The Ninja 300R doesn't have the routing for the brake and speedometer cables, but that likely wouldn't affect fender fitment. The left fork has obviously changed due to no longer mounting a speedometer gear but, eyeballing the diagram, the lowest fender stay mounting point looks higher than on J-series. It probably wouldn't be needed (especially with the two bolts on top), but the unused mount point might look funny even if you trim the parts of the stay that protrudes down to it. Eyeballing the fender, it seems that the fins don't protrude as far down the forks as they once did. It makes sense because those fins are supposed to protect USD fork tubes, but the look still went a long way to making it look like a supersport. Even if we don't have USD forks, I actually like the J-series fender design better anyway.
The upper triple clamp is the same part number (44039-0056-18R). Unless the fairings, windscreen, gauges, or inner shroud changed enough to interfere, it should fit any related clip-ons or accessories from the J-series.
Brake levers are exactly the same part (46092-0028). The entire cylinder, reservoir, lever, etc are exactly the same (43015-0110).
Front axle is the same (41068-0024) but spacers and forks are likely different due to elimination of speedometer gear and new brake setup, so don't expect a straight swap unless swapping forks and using an alternative speed pickup (Koso gauge magnetic pickup, for example). Rear axle is also exactly the same (41068-1223). That alone certainly won't tell us if that means the rear tire will bolt on with no changes, but you'd want a matching front wheel anyway.
As expected, the left and right handlebars (39058-0007-18R and 39058-0008-18R) are identical.
Strangely, the left-hand grip part number has changed from 46075-1144 to 46075-0564 despite the style looking the same and it obviously going on the same handlebars. The right-hand control housing/assembly is the exact same (46091-0205) but the throttle assembly has changed from 46019-0001 on the EX250J to 46019-0060 on the EX300. My guess is that the grip length changed slightly. That shouldn't matter if you slide some things around on the handlebars. Worst case scenario is spacing your bar-ends, getting new bar ends, or drilling new holes for the pegs in the controls and perches.
The clutch lever *has* changed from 46092-0030 to 46092-0043, but the perch and pivot bolts stay the same (13280-0295 and 92154-0066), so it's almost guaranteed to fit. It could be something as trivial as it now being knurled, painted differently, shaped with finger indentations or the length could be ever so slightly different to match the theorized grip length changes. Remember: The left-hand control housing should use less space due to no longer needing a choke lever. I'm actually curious now if these are the same part numbers from the EX250K (FI/Euro-spec Ninja 250R).
The owner's tool kit now comes with one less allen wrench, one less Phillips blade (no #3PH), and no spark plug tool (16MM).
Kawasaki shows Dunlop Arrowmax GT501 tires for the 2012 model EX250, which was only stock on some 2008 bikes so I can't confirm the part number difference for the 110/70/17 front tire (reviewers say Kawi claims tire was reformulated or made stickier). In case anyone else can find the Kawi part number for the IRC RoadWinner front 110/70/17 front on the EX250, the EX300 part number is 41009-0514.
In case anyone was wondering if their 300 will need a new shim kit for valve jobs: valve shims have the same part numbers for the same thicknesses, so they are the same diameter. No, I'm not going to list the part numbers for each size.