The statement you want to prove is true, and your argument works - but there's a simpler one: drop all reference to the recursion theorem and Godelian incompleteness, and just change the second clause in your definition of $\psi_e(n)$ to "$\sigma(n)\wedge\exists x(x\not=x)$." That is, introduce an inconsistency directly rather than playing with deeper facts about the theory and computability.

As you say, the only way $W_\theta$ can be consistent is if that second case doesn't occur, and if $\forall x\theta(x)$ is indeed true then that second case never occurs and $W_\theta$ "is" PA.

Given the directness of this construction, I suspect that there is no explicit reference for this fact.

The statement you want to prove is true, and your argument works - but there's a simpler one: drop all reference to the recursion theorem and Godelian incompleteness, and just change the second clause in your definition of $\varphi_e(n)$ to "$\sigma(n)\wedge\exists x(x\not=x)$." That is, introduce an inconsistency directly rather than playing with deeper facts about the theory and computability.

As you say, the only way $W_e$ can be consistent is if that second case doesn't occur, and if $\forall x\phi(x)$ is indeed true then that second case never occurs and $W_e$ "is" PA.

Given the directness of this construction, I suspect that there is no explicit reference.