This Microsoft MVP Access blog seems to answer my question in that there does not seem to be a programmatic way to forget/clear the connection cache.
Even so, here are three options for others to pursue, though I encourage others to read the blog first.
The conditional MS Access policy time out needs to be expanded. Although the security team isn't really budging on this, I still think this is the first solution to explore for anyone else facing my problem.
Using a SQL authenticated account. This is not best practice in my opinion. You can create a dedicated SQL Server authenticated account (or several) and share this with account and the password with users. The obvious risks are outlined more clearly in the MS blog.
Create a custom library to pass the refresh token. The way I might do this is use a .NET (with COM) approach and build something using the MSAL library. Then integrate this into VBA and update the connection string to use the token using the SQL_COPT_SS_ACCESS_TOKEN attribute.