Is it possible to use an API call to share a folder with a user using the folder path instead of the folder id?
No, /2/sharing/add_folder_member is the only way to invite a new member to a shared folder, and it only accepts a shared folder ID, not a path.
You can get the shared folder ID in any Metadata object for the folder, e.g., as returned by /2/files/get_metadata, which does accept a path.
When initially creating a shared folder, or making an existing folder a shared folder, using /2/sharing/share_folder, you must supply a path.
When inviting another user to a shared folder though, using /2/sharing/add_folder_member, you need to supply the shared folder ID.