A proof of the aforementioned result can also be found in the text of Kalton, Peck, and Roberts(p.75). While working on something else, I think I have stumbled on a general principle that would prove the same conclusion without the assumption of separability.
This is a very interesting question.
Followers of this thread may find the following papers interesting:
A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics
https://www.researchgate.net/publication/51891309_A_Birkhoff_type_transitivity_theorem_for_non-separable_completelymetrizable_spaces_with_applications_to_Linear_Dynamics
On (mk)-hypercyclicity criterion
https://www.researchgate.net/publication/283228261_On_mk-hypercyclicity_criterion
Hypercyclic and mixing operator semigroups
https://arxiv.org/pdf/1209.0974.pdf
For non-separable F spaces, see
Subseries convergence in topological groups and vector spaces
https://www.researchgate.net/publication/225555292_Subseries_convergence_in_topological_groups_and_vector_spaces
On the Orlicz-Pettis property in nonlocally convex $F$-spaces
https://www.researchgate.net/publication/303491779_On_the_Orlicz-Pettis_property_in_nonlocally_convex_F-spaces
Article A Birkhoff type transitivity theorem for non-separable compl...
Article On (mk)-hypercyclicity criterion
Article Subseries convergence in topological groups and vector spaces
Article On the Orlicz-Pettis property in nonlocally convex F-spaces
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