In addition to weakening the logic (or avoiding the parts that introduce incompleteness, especially the existential quantifier) you also need to weaken your arithmetic. We know that Primitive Recursion (PR) is too weak to prove its own consistency, and Peano Arithmetic (PA) is too strong. So you may have to look at a strengthening of PR and/or weakening of PA in order to achieve a theory that proves its own consistency. Whether that theory exists is still an open question.