The Mallows measures on the hyperoctahedral group

The Mallows measures on the symmetric group $S_n$ form a deformation of the uniform distribution. These measures are commonly used in mathematical statistics, and in recent years they found applications in other areas of mathematics as well.
As shown by Gnedin and Olshanski, there exists an analog of the Mallows measure on the infinite symmetric group. These new measures are diffuse, and they are quasi-invariant with respect to the two-sided action of a countable dense subgroup.
The purpose of the present note is to extend the Gnedin–Olshanski construction to the infinite hyperoctahedral group. Along the way, we obtain some results for the Mallows measures on finite hyperoctahedral groups, which may be of independent interest.