Catalyst Diagnosis
P0420 indicates the oxygen sensors are detecting excess O2 sensor activity downstream relative to the corresponding upstream O2 sensor. This is how it interprets a cat problem. This system uses a multiple drive-cycle verification before it runs the catalyst self-test explaining why you're not getting an error right away. It also has to fail two to three times in a row in order to trigger the Light. If it passes several times it resets the counter. Usually an intake leak won't trigger a cat code and you'd see a lean code P0171 or P0174. An exhaust leak can trigger it as air entering above primary causes the cat to overheat and could have damaged it over a long time. Disconnecting the battery is the same thing as resetting the fault codes. The ECM has to re-run its drive-cycle protocol to get back to the catalyst test.
First, do you hear an exhaust leak? cyclic tick from the driver's side, usually worse when the engine is cold and can lessen as it warms up?
Second, Hows the engine running? Can I assume the new engine got nice new double platinum Champion/Mopar spark plugs (OEM not G3 or some ridiculous mega-tipped fad plug). If memory serves you have the solid coil-over-plug distributorless setup.
The cat could have been damaged by the old engine and just now showing up.
Couple things you can do.
Down and dirty after rapping on the cat to see if it rattles, digital thermometers are cheap. You can check inlet/outlet temps. Decent ones (less than $20 on Amazon) have a laser pointer. Should run engine at 2000 rpm for about 3 minutes to fire up the cats then check inlet/outlet temps. Should see a 250-300 deg. F increase on outlet but sometimes its precarious to get in there accurately so easiest is to compare bank one to bank 2. If the suspect outlet is cooler, yup, you've got a bad cat.
More technical but more accurate, if you have a semi-decent scan tool at your disposal with data, you should easily be able to look at bank 1 and 2 O2 sensor data. B1S1 and B1S1 (the upstream sensors) should be switching relatively quickly like every second and at a range of about a volt dc. (e.g. .09 - .9) and B1S2 and B2S2 should be holding for 5 - 10 seconds at rich/lean before switching. This indicates the cat is doing its job by burning excess fuel which in turn consumes the relative oxygen. The lack of oxygen post-cat indicates the cat is working as designed. Again, let me stress, you have to get the cats hot to get them firing by running the motor at 2000-3000 rpm for 2-3 minutes.
If you see faster switching on one downstream cat but it is still significantly slower than the relative upstream cat (both same bank) then you might be able to bring it back to life with a JB emissions service. This service uses a methanol base which will superheat the cat and can burn off oil and fuel contaminates that are limiting the cats efficiency. I been using this service in my shop for the last seven years with about 50/50 results salvaging not-dead cats. Good news is its a hugely beneficial service even if the cat doesn't survive.
More info on the scan tool. While you're in there, check fuel trims. If you haven't jumped your battery or cleared codes lately, you can check after your cat warmup at idle. If you have cleared stuff, you'll want to take a few mile drive first. The fuel trims short term and long term indicate the lean/rich parameter the ECM is using to control fuel. You want to see 0 +/- 5. Ideally, a good running engine typically holds +/-2 but a stroker with stock t-body and injectors might not be able to. If you see STFT +10 or more you've got an issue. Take the engine up to 2000 rpm and see if the trims level out heading back near 0. If so, you've likely got a vacuum leak. If it stays + then you've likely got a fuel delivery issue. This will not cause a cat code directly, but the + indicates adding fuel to compensate for a lean condition. This addition of fuel over time will waste a cat.
Hope this helps.