In general, states with total angular momentum $J$ obtained from the triple couplings $j_1j_2j_3$ can be obtained in a number of ways.
A first is to couple $j_1j_2$ to $j_{12}$, and couple this to $j_3$ to get $j$, giving states symbolically denoted by $\vert (j_1j_2)j_{12}j_3;jM\rangle$. A second is to couple first $j_2j_3$ to $j_{23}$, and then $j_1$ to this to get $j$, with states denoted $\vert j_1(j_2j_3)j_{23};jM\rangle$.
The states $\vert j_1(j_2j_3)j_{23};JM\rangle$ are not linearly independent from the $\vert (j_1j_2)j_{12}j_3;JM\rangle$. In fact, the overlap is related to a $6j$ coefficient:
$$
\langle (j_1j_2)j_{12}j_3;jM\vert j_1(j_2j_3)j_{23};jM\rangle
=(-1)^{j_1+j_2+j_3+J} \sqrt{(2j_{12}+1)(2j_{23}+1)}
\left\{\begin{array}{ccc}
j_1&j_2&j_{12}\\
j_3&j&j_{23}\end{array}\right\}\, .
$$
and the overlap need not be $\pm 1$.
Your specific case seem to have $j_{12}=j_{23}=J$. I think you should follow that route to understand the effects of your permutation $j_1\leftrightarrow j_3$, i.e. write states in $\vert (j_1j_2)Jj_3;jm\rangle$ basis, with CG and all, permute $j_1$ and $j_3$, and then take the overlap of the resulting state $P_{13}\vert (j_1j_2)Jj_3;jm\rangle$ with your original basis $\vert (j_1j_2)Jj_3;jm\rangle$ to get the effect of the permutation. There are equalities of the type
$$
\sum_{e\epsilon} (-1)^{2e}\sqrt{(2c+1)(2d+1)} C_{b\beta;d\delta}^{e\epsilon}C^{e\epsilon}_{f\varphi;c\gamma}
\left\{\begin{array}{ccc}
a&b&c\\
e&f&d\end{array}\right\}= C_{a\alpha;b\beta}^{c\gamma}C_{a\alpha;f\varphi}^{d\delta}
$$
which can be found in
Varshalovich, D.A., Moskalev, A.N. and Khersonskii, V.K.M., 1988. Quantum theory of angular momentum.
to help you sort this out. The most explicit example of this procedure I could find is in this paper:
Rowe, D.J., Sanders, B.C. and De Guise, H., 1999. Representations of the Weyl group and Wigner functions for SU (3). Journal of Mathematical Physics, 40(7), pp.3604-3615.
where the effect of permutations is examined, although in a slightly different context.
