One can get a distribution where the PDF matches that of InverseChiSquareDistribution[v, s] but I'm not seeing how to get Mathematica to recognize that distribution as a scaled inverse chisquare distribution.

Here is some code to get that distribution and compare the pdf to an inverse chisquare distribution:

distY = TransformedDistribution[1/u, u \[Distributed] ChiSquareDistribution[v]];
distX = TransformedDistribution[s v y, y \[Distributed] distY]
(* TransformedDistribution[\[FormalX] s v, \[FormalX] \[Distributed]  InverseChiSquareDistribution[v]] *)
FullSimplify[PDF[distX, x], Assumptions -> v > 0 && s > 0]

PDF[InverseChiSquareDistribution[v, s], x]