Cast iron tub with hidden stopper down the overflow tube.

The stopper was jammed (probably from lack of use; I mostly take showers).
I tried loosening with penetrating oil (Lime-Away, too) but to no avail and
now the reach rod has broken free of the stopper from pulling too hard. I
believe the normal thing would be to reach down with a coat hanger wire
formed into a hook but that's not going to be effective with a stuck
stopper. How can I loosen it so that has a chance of working?

I'm thinking of running HCl down from the overflow to eat up any scale but
the stopper itself is (if I recall) brass so this might make things worse.
The stopper and passage way is going to have to be very loose for it to be
picked up with a hook.