Is there a way to encrypt a folder and then only allow certain processes to have access to that folder?
As far as I know, there is no such software available on the net. You can turn on EFS encryption for your private folder, but Windows doesn't give you options to grant access rights for different processes, and EFS only works on NTFS partitions.

There is a third-party software called Lock My Folders, which works the similar way like EFS, and it lets you create a secure folder to transparently encrypt/decrypt the files inside. But it also lacks the feature of restricting process access to the encrypted folder.