This question extends my inquiry from a previous post .
Dan Willard's Self-Justifying Axiom Systems/Self-Verifying Theories  and Brown and Palsberg's self-interpreter for F-Omega  both employ techniques that might be described as "excluding diagonalizaton" from a given domain, while preserving other properties usually considered as concomitant with diagonalization.
Willard: "In outline, the key to Willard's construction of his system is to formalise enough of the Gödel machinery to talk about provability internally without being able to formalise diagonalisation. Diagonalisation depends upon being able to prove that multiplication is a total function (and in the earlier versions of the result, addition also). Addition and multiplication are not function symbols of Willard's language; instead, subtraction and division are, with the addition and multiplication predicates being defined in terms of these. Here, one cannot prove the PI-0-2 sentence expressing totality of multiplication:"
Brown and Palsberg: "After a careful analysis of the classical theorem, we show that static type checking in Fω can exclude the proof’s diagonalization gadget, leaving open the possibility for a self-interpreter."
Therefore: Are these techniques related?
Separate from this core question, but possibly useful for answerers, I speculate that, if so, the relation might be characterizable via a common bypassing of Lawvere's Fixed Point Theorem, which itself unifies a number of diagonalization based proofs.