OK.I suggest you not do anything radical.
If you have a local gunsmith,he has likely solved this before.
If I may ask,how far into the bore is it?It matters for a couple of reasons.
If it is six or eight in? It went that far before it stopped.Is it stuck in the throat?
You might be able to use a soda straw or brass telescope tube to help you fish and fiddle the cleaning rod back onto the tip.The thin sleeve may help center the tip of your male thread.You may even be able to sleeve the rod through it.
Take the barreled receiver out of the stock.
Reversing wedges things tighter.It has to go the direction you started.
While I agree soaking something into the patch may make it swell,lubing the surface you have to travel over will help.
So you might use a.22cal size patch with some RIG grease,or case lube,to lightly coat the bore down to the obstruction.Don't use anything that could get stuck.
OK,bore is greased to the muzzle.
If there is just no way you can rethread your tip on,use a1/4 in brassrod,one piece.
Point the muzzle up,and with the rod inserted in the breech end of the barreled receiver,bump it on the floor.
Once it beaks loose and hits the grease,you ought to be able to push it through.We hope!
One more thought.If you think you may end up taking it to a smith,eventually...Stop now.