Do you know an electrician? 
I have a pice of fiberglass rod used for “fishing” electrical wires. Mine has no metal on the ends. And I’ve used it a lot when I double feed my Benjamin Marauder pistol. If you know an electrician he may give you a pice. Here is a photo of the product I’m referring to. 
1577476696_7511983715e0662580b4e06.08200384_091A848B-823B-436C-9956-919BE7E4C1ED.png

 
To the OP, I don't know your history with the Patchworm, but I had a similar experience. I tried to pull a dry patch through with the white .22 button attached, big mistake. I was trying to do a single, dry pull-through to see if it would restore accuracy during a session. Although I can do this with a Bore Snake, the dry patch worm stuck. I managed to pull it out after wrapping the line around 1" wood dowel, with the rifle secured in a vice. My lesson learned, if running a dry patch through a dirty barrel, always do it first with no button attached. With lube (I use Ballistol or WD-40), I've never had one stick. Trying to shoot it out may be fine, but something about that bothers me. The air has to go somewhere, don't know if it would blow out the breech seal, or potentially do any valve damage by going back through the transfer port. If it's blocking the TP, this is probably not a good option. Although the warning about a metal rod is wise, it could be your best option. Seems like, if the barrel is removed and well supported in a vice, you could insert a rod and tap it through. If you have one of the cone-shaped muzzle protectors that slide on the rod, that would help keep it centered, but you'll need a third hand. If kept centered, I doubt it would flex enough to damage the barrel. Ordinarily, I would do this from the breech, but give the patch location, probably better doing it from the muzzle. If you are careful with its insertion, you shouldn't do any crown damage. Good luck. 
 
You can get the fiberglass rods at harbor freight they will not break as easy as a wood dowl that size look in electrical 

I would not try shooting it out because if it stuck hard enough for you not to push it out it most likely will not shoot out 

as suggested use lube stay calm an keep working on it pushing it with a rod or dowl, a lot of times wood dowels will have a knot in them that will cause them to break at the weak point 

the fiberglass rod is the best option

Instead of running the rod down to the plug, pull it out 1/2 to 3/4 of the barrel and then fling it down the barrel letting it STRIKE the plug like a hammer this snapping creates more force you will need to do this multiple times but there is less chance of the rod breaking and it will work the plug out