Hi,
There is no application note regarding your requirement.
This is common use case. You boot your device on core0 and prepare everything for safety critical application execution. (perform self-tets, calibration, etc..)
Once all of this is done you will start core1 in lockstep by your SW (refer to reference manual SIU chapter) where all your safety related tasks will run and put core0 into reset from core1.
from this point on, you will execute only in lockstep.
You can find useful our example codes:
https://community.nxp.com/docs/DOC-329623#MPC5777C
regards,
Peter