Yes, your conjecture is true. You can prove this as follows. Let $\omega$ be an $SU(2)$-invariant Kahler metric of $X$.

1. Show that at any point of $X$ the tangent spaces at both factors $\mathbb{P}^1$ and $\Sigma$ are perpendicular.

As a corollary there are smooth functions $f,g$ on $X$ such that $$ \omega = f \, \pi_2^* \omega_1 + g \, \pi_2^* \omega_{FS}$$ where $\omega_1$ is a Kahler metric on $\Sigma$

2. Show that $f,g$ are constant by using that $\mathrm{d} \omega = 0 $ implies $\mathrm{d} f = \mathrm{d} g = 0$. This is a local computation by using local coordinates of $\Sigma$ and $\mathbb{P}^1$.

Let me know if need more details about 1) or 2).

Yes, your conjecture is true. You can prove this as follows. Let $\omega$ be an $SU(2)$-invariant Kahler metric of $X$.

1. Show that at any point of $X$ the tangent spaces at both factors $\mathbb{P}^1$ and $\Sigma$ are perpendicular.

As a corollary there are smooth functions $f,g$ on $X$ such that $$ \omega = f \, \pi_1^* \omega_1 + g \, \pi_2^* \omega_{FS}$$ where $\omega_1$ is any Kahler metric on $\Sigma$.

2. Show that $g$ is constant and that $f = \pi_1^* h$ is the pullback of a smooth function $h$ on $\Sigma$. Use that $\mathrm{d} \omega = 0 $ and the $SU(2)$-invariance of $\omega$. This is a local computation by using local coordinates of $\Sigma$ and $\mathbb{P}^1$.

3. Now $\omega = \pi_1^* \omega_2 + a \, \pi_2^* \omega_{FS} $ as you conjectured, with $\omega_2 = h \omega_1$ which is a Kahler metric on $\Sigma$.

Let me know if need more details about 1) or 2).