I'm encountering an odd issue with one particular SQL Server database. This database is not managed by me, it is for another application, but I have read access to a handful of tables in the DB. I can connect to the database just fine from ArcGIS Desktop and SQL Server Client, but if I attempt to connect to the same database from Pro using the same credentials on the same machine, it fails. The general error message is that a database connection exception has occurred:

I don't have this issue with any of our other databases. The only difference I've noted from the DBs that I don't have this issue with is that it lives on a different server machine. Both machines are running SQL Server 17. I asked IT to check the logs on that machine, and it doesn't seem to register the connection attempt, or at least not register an error. I also asked them to confirm that the DB user has the correct permissions, which it does.
I've also made sure to download the latest 64-bit ODBC driver and tried moving it before the 32 bit version in the PATH variable, but no luck yet. Has anyone encountered an issue like this before?