What are the risks of upgrading DB major version in replica and then promoting it as master?
There are risks.  I doubt if anyone can enumerate all the risks.
Plan A:  Dump all the data via mysqldump (or other dump utility); upgrade the system; load the data.  This usually lets you safely do multi-major-version upgrades.
Plan B:  Upgrade one major version at a time.  (This has always been a requirement in MySQL, especially when going to 8.0.  MariaDB does a better job at relaxing the limitation to a single major.)
Plan C:  Get RDS to do the work.  I would expect that they have well-tested upgrade paths for when they force an upgrade.  (The Question is unclear as to whether your 10.1 is currently hosted by AWS or yourself.)
I would go with 'C' since they have all the tools to do everything with minimal downtime, etc.  They will probably use snapshots, etc, which you don't have easy access to.