Administrative recognition of frequently-cited unpublished manuscripts
how do you convince administrators that ...


You don't and you can't, and I endorse Alexandros's suggestion.

The department to which you have applied, if they want to hire you, will try to convince the administrators of exactly this. The admins don't know you and don't trust you, but they might listen to such complaints coming from someone they trust.

In the meantime, in addition to any other benefits, submitting your paper to a peer-reviewed journal will signal to the hiring committee that you are doing your best to make their lives easy.