I'd speculate that these are the linear systems on the Kummer surface given by twice the curves described in Hudson's "Kummer's quartic surface" sections 90 and 91: identifying the Kummer surface with it's image in $\mathbb{P}^3$ (as a quartic surface with 16 nodes at the images of the 2-torsion points), and choosing a 6-tuple of nodes which sit on a plane, there is exactly one reduced sextic on the Kummer surface which passes only on this 6-tuple of nodes out of the 16, and exactly one sextic on the Kummer surfaces which pass only on the residual 10 nodes out of the 16. The numerology (degrees of these divisors, their uniqueness, the nodes they pass through) suggests that these are halves of the linear systems in question.

I'd speculate that these are the linear systems on the Kummer surface given by twice the curves described in Hudson's "Kummer's quartic surface" sections 90 and 91: identifying the Kummer surface with it's image in $\mathbb{P}^3$ (as a quartic surface with 16 nodes at the images of the 2-torsion points), and choosing a 6-tuple of nodes which sit on a plane, there is exactly one reduced sextic on the Kummer surface which passes only on this 6-tuple of nodes out of the 16, and exactly one sextic on the Kummer surfaces which pass only on the residual 10 nodes out of the 16. The numerology (degrees of these divisors, their uniqueness, the nodes they pass through) suggests that these are halves of the linear systems in question.

Edit:

I went back to Hudson and read the actual section (instead of writing from memory - bad habit): there is a 5-dimensional family of sextics through these 6 nodes, and a 4-dimensional family of sextics through the residual ten. The corresponding linear systems are 4 and 3 dimensional respectively, they are both invariant under the kummer involution, and are non intersecting subspaces $|3\Theta|\cong\mathbb{P}^8$ (and therefor span it). It is also easy to check that they are base point free. I think this is closer to being a complete answer to the question

I'd speculate that these are the linear systems on the Kummer surface given by the curves described in Hudson's "Kummer's quartic surface" sections 90 and 91: identifying the Kummer surface with it's image in $\mathbb{P}^3$ (as a quartic surface with 16 nodes at the images of the 2-torsion points), and choosing a 6-tuple of nodes which sit on a plane, there is exactly one reduced sextic on the Kummer surface which passes only on this 6-tuple of nodes out of the 16, and exactly one sextic on the Kummer surfaces which pass only on the residual 10 nodes out of the 16. The numerology (degrees of these divisors, their uniqueness, the nodes they pass through) suggests that these are the linear systems in question.

I'd speculate that these are the linear systems on the Kummer surface given by twice the curves described in Hudson's "Kummer's quartic surface" sections 90 and 91: identifying the Kummer surface with it's image in $\mathbb{P}^3$ (as a quartic surface with 16 nodes at the images of the 2-torsion points), and choosing a 6-tuple of nodes which sit on a plane, there is exactly one reduced sextic on the Kummer surface which passes only on this 6-tuple of nodes out of the 16, and exactly one sextic on the Kummer surfaces which pass only on the residual 10 nodes out of the 16. The numerology (degrees of these divisors, their uniqueness, the nodes they pass through) suggests that these are halves of the linear systems in question.