Assuming there’s no technical issue like @kevistraining suggested: In my experience, measurement discrepancies between power measuring devices (trainers, power meters, whatever) are not consistent at every wattage. I’ve had a trainer and spider that were identical at around Z2 watts, and a good 15%+ off each other at threshold watts. I’ve also had a trainer and spider that track super well. The marketing materials on everything has insisted they’re all accurate to within like 2.5%, it’s all nonsense 
All your power measuring systems are yardsticks of slightly differing lengths. You have a new yardstick, so your best option is use only the new yardstick. Forget whatever your Kickr Core said and embrace the pedals.
TR doesn’t support multiple power profiles, but AT will get you to the right place after a few rides. If you want to speed up the process, choose harder alternates until they feel “right” (or until you start failing, that works too).
