Hi Kaylee,
It's not possible to selectively restrict certain statuses to certain users. The only functionality you have available is to customise the statuses that are available to different groups, however the entire list of statuses available to the group associated with your project will also be available to requesters, which if i'm understanding correctly isn't going to be suitable.
The only alternative I can think of is to trigger your automation from a custom field instead of a status. Going down this route would enable you to hide the form/field to the select few users who you want to be able to trigger the automation.
Best Regards,
Rich