Tarmo Uustalu <tarmo@cs.ioc.ee> has informed me that "It is not true
that Rauszer didn't study sequent systems for H-B logic. In her paper
[1], she proposes a Gentzen system where cut is eliminable (she
claims, I haven't checked the proof). Her condition on sequents is: in
no sequent may the antecedent and succedent both consist of more than
one formula."

[1] C Rauszer. A formalization of the propositional calculus of H-B
logic. Studia Logica 33(1):23--34, 1974.

Tarmo has now found a counter-example to the cut-elimination theorem
of Rauszer. Details can be found in:

A Cut-free Sequent Calculus for Bi-Intuitionistic Logic 
Linda Buisman and Rajeev Goré 
Proceedings of the International Conference on Theorem
Proving with Analytic Tableaux and Related Methods Aix en Provence,
France, July 2007, LNAI 4548:90-106, Springer-Verlag, 2007.