Actually, I did an "opam update" (after my comment that stated there was "3.7.0") and "opam show dune --all-versions" no longer shows "3.7.0". :|

It shows:

  all-versions         1.6.3  1.9.3  1.11.4  2.3.0  2.4.0  2.5.1  2.6.1  2.7.1
                       2.9.3  3.5.0  3.6.2  3.10.0  3.12.1  3.12.2  3.15.3
                       3.17.2  3.18.2  3.19.0  3.19.1

Your alternative really is:

  $ opam pin add dune https://github.com/ocaml/dune/releases/download/3.7.0/dune-3.7.0.tbz

Just FWIW, maybe it might work the way you wanted it to if you do not update the registry; worth a try. It should be able to fetch the correct tarball.