Relationship between different Gödel Sentences

First note that your alternative shifting numbering scheme is acceptable. For example, the number of a wff $P_1 \lor P_2$ ($2^{361}×3^{7}×5^{529}$) under your specific new numbering can be converted to $2^{289}×3^{9}×5^{361}$ under the original Gödel numbering, and any such conversion function must be p.r. too. In other words, you can program in advance to convert any given number under your numbering to Gödel number without open searches due to the fundamental theorem of arithmetic and your specific deterministic simple shifting scheme.

Any Gödel sentence (at least one such sentence exists per diagonal lemma) in PA has the following property: $G↔ \lnot Bew(\ulcorner G \urcorner)$ where $\ulcorner G \urcorner$ is the usual Gödel number of a Gödel sentence, $Bew(y)=∃x(y=\ulcorner \phi \urcorner \land x=\ulcorner \psi \urcorner)$, and $\psi$ is a proof of a sentence $\phi$, and the name $Bew$ is short for beweisbar, the German word for "provable" and was originally used by Gödel to denote the provability formula above (see reference). However, it's a well known result that the numerical provability property $Bew(y)$ is not p.r., otherwise any first order sentence $\phi$ whose Gödel number is $y$ can be p.r. decidable contradicting the first incompleteness theorem. Given $G_0$ is the Gödel sentence associated with Gödel's original scheme and suppose $G_1$ is a Gödel sentence associated with your alternative scheme, of course we have $PA \cup \{G_0\} \vdash G_0 \land Bew(\ulcorner G_0 \urcorner)$ trivially. Although we can always find a p.r. function $f$ converts $\ulcorner G_1 \urcorner$ under your shifting scheme to a number under original Gödel scheme, any recursively axiomatized theory containing PA cannot decide the truth value of the provability predicate $Bew(f(\ulcorner G_1 \urcorner))$ even after adjoining $G_0$ to PA since there's no known theorem to always ensure $f(\ulcorner G_1 \urcorner) = \ulcorner G_0 \urcorner$ and the numeral value of $f(\ulcorner G_1 \urcorner)$ can be arbitrary. So similarly under your description in general $PA \cup \{G_m\} \nvdash G_n$ (the augmented theory can not capture $G_n$).