Only a partial answer for the special case of an integer $n$ with Galois radius $r$ of type $(a,b)$ where $\min(a,b)=1$.
Denote by $\mathbb{P}_{k}$ the set of $k$-th powers of primes and suppose $b>a$, so that $(n-r,n+r)=(p,q^{b})$ with $p$ and $q$ odd primes. Provided $n>5^{b}$, consider the closest odd element $q'$ of $\mathbb{P}_{b}$ to $p$ less than $p$ and set $2s=p-q'$. Then $s$ fulfills the required conditions to be an inversion shift of $r$. The same line of reasoning applies mutatis mutandis for the case $b<a$.