Hypersequents: proof term assigments or translations to hybrid logic
I know this is a bit late, but perhaps it is still of interest.

While not exactly the logic you are interested in, Gödel-Dummett logic, the intermediate logic characterised by linear Kripke-frames, is closely related to S4.3, and for this logic people have looked into similar questions:


The paper A Lambda Calculus for Goedel-Dummett Logic Capturing Watifreedom by Yoichi Hirai gives a lambda calculus based on a hypersequent calculus for that logic.
The paper A Natural Deduction System for Intuitionistic Fuzzy Logic by Matthias Baaz, Agata Ciabattoni and Christian Fermüller gives a natural deduction system for the same logic, which might be used to give proof term assignments.


Also, you might be interested in the paper From Frame Properties to Hypersequent Rules in Modal Logics by Ori Lahav, which includes an alternative hypersequent calculus for S4.3.