We moved from 2nd grade private to 3rd grade AAP this past fall for a few reasons
1) Private was only a covid bandaid while both of us still had to WOH. It wasn't ever a long term plan
2) The private school admitted to us they couldn't/wouldn't meet any advanced academic needs, that they were focused on grade level material only.
3) His best friends from private school were all leaving for AAP as well (a lot of other covid families)
We applied in the fall of his 2nd grade year and got a response in winter, in time to decide not to sign a contract at the private for the following year.
I'm not sure why differing curriculums would be a problem at a young age. As long as your kid can read/write/do basic arithmetic, they're on track for 3rd grade regardless of how they learned it. We had no issues.