I had a Johnathon Browning rifle and my breech came out easy peasy.
Were it me however ... I would continue with the "shooting it out" program till it comes out. Getting enough powder in there will require 3F or 4F powder. You will not get too much powder in there gently shoving it in with a bent wire. It will however need to be dry behind your rod tip. This will dislodge it with perseverance. I can not understand what damage this could cause. I have done this also a few times with jag's, rod tips, and dry balls that I could not pull with a screw tip.
I hate those hollow breech plugs. A good flat end plug is the best in any world that I can think of. No muss ... no fuss! Unless you set the barrel back tho ... you are stuck with your breech plug. If you ever do remove the plug however, I would recomend smoothing that lil chamber to mirror bright using 150 sand paper followed with 220 grit then 400 grit. If a brighter finish is desired then follow that with 800 then 1500 grit. You will thank yourself with the ease of cleaning and reduced chance of sticking another rod tip