I would love to use monero to "power" my private block chain app. Would this be overkill? Would ethereum be secure "enough"?
Either Monero or Ethereum would be overkill. Saving non-financial data on Monero would be expensive. What you're looking for is an encrypted database.
Monero is private in the sense that users are anonymous. Private blockchains in the sense that they're run by one entity are simply inefficient databases. The security and immutability of a blockchain depends on having multiple users run nodes and having multiple miners create blocks.
In the case of school transcripts, the school can be trusted to keep its own records and a trustless solution will not be necessary.