What is the reason the users cannot have access to that field if it is only for the status?
Are there any other actions that are performed on the record when that status needs to be set that you may be able to use as a trigger for a form rule?
I don't know if this is best practice in this case but you could have a checkbox(can get annoying on forms) that is only shown to users in the roles without access to the Status field. An automation could watch for this box to be checked and when it is it could perform the action you want of changing the status.