Is checking data integrity with hashes useful for my scenario?

You need to be very careful with the user data checksumming! For example, enabling checksumming for ReFS turns it into fully log-structured file system with all the I/O patterns changed.

https://www.starwindsoftware.com/blog/log-structured-file-systems-microsoft-refs-v2-investigation-part-1

Your database app won't like it for sure as DBs use log in front of the "flat" storage to accelerate writes, with ReFS you'll end up with log-on-log concept and it's a terrible idea performance-wise.

https://www.usenix.org/system/files/conference/inflow14/inflow14-yang.pdf

I think you can safely rely on your backup of choice handling data integrity checks for you. How much of the downtime can you afford?