Epimorphisms from the genus $2$ surface braid group to finite groups

I have found a finite image of order $3^9$ in which $\sigma$ has nontrivial image, by using the $\mathtt{pQuotient}$ function which, for a given prime $p$ and class $n$, computes the largest quotient of the group which is a $p$-group of exponent $p$-class at most $n$.

gap> pq := PQuotient( Br, 3,2);
  <3-quotient system of 3-class 2 with 9 generators>
gap> phi := EpimorphismQuotientSystem(pq);
  [ a1, b1, a2, b2, s ] -> [ a1, a2, a3, a4, a5 ]
gap> Image(phi, Br.5);
  a5
gap> Order(Image(phi, Br.5));
  3

In fact this quotient has elementary abelian centre of order $3^5$, so there are certainly solutions to your problem of size $3^5$, and perhaps smaller. But factoring out a normal subgroup of order $3^4$ that lies in the centre and does not contain the image of $\sigma$ gives an extraspecial group of order $3^5$.

Consider $C_2 \times C_2$ minus the diagonal, this is a double cover of your space. Given a symplectic form on $H_1( C_2 \times C_2 - \Delta, \mathbb Z/p)$, we can form an associated extension $1 \to G \to H_1( C_2 \times C_2, \mathbb Z/p)$ as a Heisenberg group, which will be a quotient of $\pi_1$ if and only if the image of the obstruction class in $H^2 ( \pi_1(C_2 \times C_2 - \Delta), \mathbb Z/p) = H^2 ( C_2 \times C_2 - \Delta, \mathbb Z/p)$ vanishes. The obstruction class should be the class induced by viewing symplectic forms on on $H_1$ as elements of $\wedge^2 H^1$ and applying the cup product.

To ensure that $\sigma$ has order $p$ on $G$, it is sufficient to show that the obstruction class does not vanish in $H^2( C_2 \times C_2 - \Delta, \mathbb Z/p)$. In other words, we want the obstruction class to be a multiple of the class of $\Delta$. This is possible as every class in $H^2$ is a cup product.

Then because $C_2 \times C_2 - \Delta$ is a double cover of your $X - \delta$, it's fundamental group will map to the wreath product of $\mathbb Z/2$ with $G$, and probably something simpler like a semidirect product if we can choose the form to be invariant.

The minimal group size for this method to work is something like $2 \cdot 3^9$, which explains why it wasn't seen by your brute-force search.