I misunderstood your proposal, but you don’t need to do this work to get what you want. You can just take each sentence □nϕ→ϕ as an axiom, but declare that this axiom takes n symbols to invoke. This could be done by changing the notion of length of a proof, or by taking axioms ψϕ,n→(□nϕ→ϕ) and ψϕ,n with ψϕ,n very long.

I misunderstood your proposal, but you don’t need to do this work to get what you want. You can just take each sentence □nϕ→ϕ as an axiom, but declare that this axiom takes n symbols to invoke. This could be done by changing the notion of length of a proof, or by taking axioms ψϕ,n→(□nϕ→ϕ) and ψϕ,n with ψϕ,n very long.