IIUC, the solution requires rewriting `persist-copy-tree' (which is
copied from Emacs 30's `copy-tree') so that it handles hash tables.  If
you all agree, I can do this in `persist.el`.

However, would it make sense for `copy-tree' to handle hash tables?

Forgive me if this has been discussed and settled before.  I did find
this thread about comparing hash tables with `equal':

https://yhetil.org/emacs-devel/871qvz4kdw.fsf@localhost/

Thank you!

Joseph