Cody, if your breach plug is welded in, buddy, you have not one but two problems. A welded breach plug may well not be safe and it ain't right, no way. A breach plug is supposed to be screwed into the breach of the barrel. The threads give the breach plug greater strength to withstand the pressures that are generated when the gun is fired. For safety's sake, you need to have a gunsmith remove the old breach plug, thread your barrel breach and replace the present plug with one that screws in. Sure it will cost a few bucks but it sure beats having a welded breach plug come loose and go sailing back through your brain pan. That ain't a pain you will soon recover from and it will create a tough cosmetic problem for you local mortician.
Meantime, that idea about removing the touch hole liner and putting a zerk fitting into the touch hole and using a grease gun to push out the stuck ball is a wonderful and very inventive idea. Just be sure that the threads in your touch hole are compatable with available touch hole liners. It's sure be a pain to find that you had made too big of a hole or used the wrong tap to cut your threads and then have to repair your touch hole before you could replace the liner. I think he is right about tapping the hole with a 1/4-28 thread. I think most touch hole liners have that thread. Jim Chambers touch hole liners have a non-standard thread so I wouldn't order one of them even though they are wonderful liners. It would just be one more problem in matching the threads of the liner with those of a zerk fitting.
All of this assumes that you have tried the usual method of stuck ball removal by working ffffG powder into the touch hole, making sure the ball is seated and then just shooting it out. That's the usual method of removing a stuck ball if you can be sure is seated on the powder. This won't work if the ball is stuck part way down the bore. Using this method under those conditions can result in a ruptured barrel with bad consequences for you. But if the ball is seated on the powder, it is a great way to remove the ball safely.
Another bit of advice......go to your stock of gun stuff and find that bag of poly patches, carefully bundle them up so you don't drop any and run, don't walk, to your nearest trash can and
throw them away. Poly patches have no place in a flintlock. They are a modern thing intended primarily to separate the unsuspecting person from their money. They are absolutely no good in a traditional muzzleloader. What you should be using is a 100% cotton fabric that you lubricate with a non-petroleum lubricant. Good old spit is one of the best lubricants so long as you are not going to leave the gun loaded for long periods of time such as when you are hunting or it is below freezing outside. I like straight Ballistol on my patches but there are a BUNCH of alternative lubricants available along with a LOT of recipes for home made lubricants. Just DO NOT use petroleum based lubricants. Petroleum based lubricants form a gummy residue in your bore that is hard to remove and will cause you to have fouling problems.
Bottom line, use the grease fitting method to remove your stuck ball (and stuff), then take your rifle to a gunsmith and have him replace the welded breach plug with a threaded one. Lastly, get rid of the Poly Patches and use only 100% cotton fabric patches with a natural non-petroleum lubricant. Your present problems will be solved and you can then turn your attention to developing the most accurate load that your gun can deliver. For this, I highly recommend getting a copy of Dutch Schultz' muzzleloading accuracy method. You can find it by Googling "Muzzleloading Accuracy". I think Dutch is now getting $15 for his pamphlet but it is going to be the best $15 you will ever spend on muzzleloading. I just looked up his web site and it is
http://www.blackpowderrifleaccuracy.com/tips.html It involves a good bit of work to get the most accuracy from your rifle but as we ex-Marines know, an inaccurate weapon is just an expensive club.
Semper Fi!!