I don't know of logic textbooks which adopt an explicitly formalist viewpoint (which is not to say I think none exist), but for what it's worth, I'd say that books on categorical logic tend not to adopt a Platonist point of view. Abstract methods of categorical algebra naturally tend to promote the idea of there being different universes (e.g., toposes) in which to do mathmatics, and a seasoned category theorist is quite comfortable mentally inhabiting worlds where one is a straight-up model of ZFC, and another is a world where every function from the reals to itself is continuous, etc. The book my Mac Lane and Moerdijk on topos theory, and the book by Lambek and Scott on higher-order categorical logic, come to mind as examples of books which encourage this way of thinking. (The Lambek-Scott text has a bit of philosophy where they suggest possible bridges between platonism, formalism, and intuitionism.)

Probably books on categorical logic are not what you're looking for, but in that case you might try seeking online lecture notes on more traditional topics written by categorical logicians, who seem to have a habit of looking at metatheoretic contexts. One example I bumped into recently are Jaap van Oosten's notes on Peano arithmetic and Gödel incompleteness phenomena. Again, it's not that an avowedly formalist position is declared; it's more the style of writing that eschews appeals to Platonic intuitions about the "real" natural numbers (which I don't care for either). If you make your question more specific about exactly which topics you want to see, maybe a more specific answer can be made.