this is a very specific question, I recommend you to enable bpbkar logging on the Windows client and try several tests with files with special characters in their name.
Ideally speaking *.bak should be good enough to ensure that the file is excluded and the $ symbol should not really interfere with it.
Could you run the below on the destination machine to share the exclusion list
reg query HKLM\SOFTWARE\VERITAS\NetBackup\CurrentVersion\Config\ /v Exclude
If the exclusion list appears to be fine then we may need to go with the bpbkar logs on the client machine to determine why the file is not getting excluded.