is it possible to access certain models on the page with permissions that differ from the page user?

I need to be able to render a button based on a field on our object that not all users have read access to. The problem is that I need both options (no read access with button access, no read access no button access either). This means that I cannot just add ‘undefined’ to my render criteria.

Is it possible to create a model that has no layout fields and queries the object with admin or similar access level? 

Yes &  No.

Not without an additional model.

Create on model on this object. It will allow you to evaluate a users’ access rights on any record.

https://developer.salesforce.com/docs/atlas.en-us.api.meta/api/sforce_api_objects_userrecordaccess.htm