A branch-tangent Killing field restricts to the bifurcation sphere

For $p\in S$, branch tangency gives $T_p\in T_p{H}_1\cap T_p{H}_2=T_{p}S$ by the transverse-intersection theorem. Since $S$ is embedded and $T$ is smooth, the restriction theorem gives a smooth field $A=T{\mathrm{\mid }}_S$ on $S$.

Fix $p\in S$ and take a slice chart $(x^1,x^2,x^3,x^4)$ around $p$ with $S$ given by $x^3=x^4=0$. Tangential indices $a,b,c$ range over $1,2$ and normal indices $\mu$ over $3,4$. On the slice, $A^a=T^a{\mathrm{\mid }}_S$ and ${\gamma }_{ab}=g_{ab}{\mathrm{\mid }}_S$. Tangency gives $T^{\mu }{\mathrm{\mid }}_S=0$, so differentiating this identically zero function in each tangential coordinate gives $({\mathrm{\partial }}_a{T}^{\mu }){\mathrm{\mid }}_S=0$.

Apply the covariant two-tensor coordinate formula first on $S$ and then in the ambient chart. With $c$ summed over $1,2$, it gives $({\mathcal{L}}_A\gamma {)}_{ab}={(T^c{\mathrm{\partial }}_c{g}_{ab}+g_{cb}{\mathrm{\partial }}_a{T}^c+g_{ac}{\mathrm{\partial }}_b{T}^c)\mid }_S={({\mathcal{L}}_{T}g{)}_{ab}\mid }_S=0$. Indeed, the additional normal-index terms in the ambient formula vanish by $T^{\mu }{\mathrm{\mid }}_S=0$ and $({\mathrm{\partial }}_a{T}^{\mu }){\mathrm{\mid }}_S=({\mathrm{\partial }}_b{T}^{\mu }){\mathrm{\mid }}_S=0$. Thus ${\mathcal{L}}_A\gamma =0$ in every such chart, and $A$ is Killing for the induced Riemannian metric. Nontriviality means $A\equiv ̸0$ and does not require $A$ to be nowhere zero.