“Diffeology” by Patrick Iglesias-Zemmour is probably the closest match.
He develops differential forms and de Rham cohomology, fiber bundles, connections, and symplectic geometry in the language of diffeological spaces, i.e., concrete sheaves of sets on the site of smooth manifolds. This book is closest in style to a conventional differential geometry textbook.
Another book is “Synthetic geometry of manifolds” by Anders Kock, which treats differential forms, Lie groups and algebras, principal bundles with connections, jets and differential operators. It has a somewhat different focus (e.g., infinitesimals and the internal language of toposes) than the previous book.
In relation to this one can also mention “Models for smooth infinitesimal analysis” by Ieke Moerdijk and Gonzalo Reyes, which covers some foundational topics in differential geometry, like differential forms.