Express the Klein bottle group $G'=\langle T,A\mid ATAT^{-1}\rangle$ as an HNN extension of $\mathbb{Z}$ as follows (using notation from Wikipedia for convenience: https://en.wikipedia.org/wiki/HNN_extension):
Let $G=\langle a,b\mid ab\rangle$ and consider subgroups $H=\langle a\rangle$ and $K=\langle b \rangle$ of $G$ (note $H\simeq K\simeq\mathbb{Z}$ by the Freiheitssatz for one-relator groups). Define an isomorphism $\alpha:H\to K$ by $\alpha(a)=b$. Then $G*_\alpha$ is an HNN-extension of $G$ with presentation $G*_\alpha=\langle t,a,b\mid ab,tat^{-1}=b\rangle$.
Note that $G*_\alpha\simeq G'$ via $t\mapsto T$, $a\mapsto A$, $b\mapsto TAT^{-1}$ (this homomorphism has inverse $A\mapsto a,T\mapsto t$).
Britton's Lemma implies that the word $w=at^{-1}at$ is nontrivial in $G*_\alpha$ since it has positive length but contains no subwords of the form $tht^{-1}$ or $t^{-1}kt$ for $h\in H$ or $k\in K$.
But $w\mapsto w'=AT^{-1}AT$ under the isomorphism between $G*_{\alpha}$ and $G'$, and $w'=(AT)^{-1}(ATAT^{-1})AT=1$ in $G'$, appearing to contradict Britton's Lemma.
I suspect this is a subtlety related to cyclic reduction. I think the problem goes away if you require subwords of the form $tht^{-1}$ or $t^{-1}kt$ to appear in some cyclic permutation of $w$ in the hypothesis of Britton's Lemma. But no source that I can find state's Britton's Lemma this way and reviewing the proof in Lyndon and Schupp (starting pg 181) hasn't helped me see where I'm going wrong.