I really wish I had a guaranteed and easy solution to your problem, but the best I can offer is a remote possibility of success. It's a good assumption that the stuck patch is not going to be pulled the rest of the way through the bore, so it needs to be pushed back out into the chamber. However, the problem is, you can't push the Bore Snake.
I've had a similar problem using an Otis cleaning cable. Like the Bore Snake, it's used to pull the cleaning patch through the bore, but unlike the BoreSnake, in the tight .223" bore, the cleaning cable can be used in compression (pushing) as well as tension (pulling). I was able to clean the oil off the cable's vinyl outer sheath to get a good two handed grip, pinch the PLR-16 between my knees while seated, and make short compressive strokes that jack hammered the patch back out into the breech.
If the .22 Bore Snake is a hollow fabric tube, it might be possible to cut it off a couple of inches beyond the muzzle and slide a small steel rod inside this sheath and tap it out. If the Bore Snake is a hollow nylon sheath around a nylon cord or small diameter rope, this probably won't work. The .223" bore doesn't leave a lot of room to work. If you have 8" of bore that's already filled with compressible Bore Snake, it's going to be difficult to tap the obstruction back out from the muzzle end.
I considered trying to using a sharp brass screw from the breech end to screw into the back side of the plug and pull it out into the chamber, but the back of the breech in the PLR isn't open, so that makes this difficult, and screwing into the back of the patch plug would compress it more tightly into the side walls of the bore, making it even more difficult to extract. However, when someone forgets to put a powder charge into a muzzle loader and rams a lead ball all the way down the bore, there is a tool that is essentially a self drilling and self tapping screw on a long rod that can be used to drill into the lead ball and grab it, so it can be pulled back out of the bore using the inertia of a sliding weight on the exposed rod, much like a slap puller. But the PLR-16's tightly enclosed receiver area makes it difficult to work on this problem from the chamber end.
It might be possible to use a short pistol length Otis cleaning rod as a flexible shaft for a drill and use a belt sander to modify one of the brass tools in the Otis cleaning kit with the 8-32 threads that screws into the brass end on the cleaning cable as a patch chewer and use a cordless drill to spin it into the back of the patch plug, to slowly chew up the back end of the patches until the plug is sufficiently eaten away that it can be pulled out the muzzle.
Barring these mechanical solutions, there may be a chemical solution that works overnight. Maybe do some research to determine what dissolves a Bore Snake but doesn't dissolve steel. Chemical disincorporation. Where's Walter White when you need him?
The main reason I don't use Bore Snakes is that I don't like pulling a dirty Bore Snake through the bore and I don't like storing a dirty solvent and crud loaded Bore Snake nor washing them in the washing machine and untangling them. I like pulling a clean patch and when it comes out the other end I toss the little dirty patch in the trash. One and done. However, you have demonstrated another good reason not to use a Bore Snake, particularly in a .22 bore. If you get one stuck, it's suddenly and unexpectedly a very big deal. The cleaning cables can be pushed back down into a .22 bore, and possibly up to a .308 bore. This has pulled my bacon out of the fire on a few occasions.
In defense of the Bore Snake... I thought they recommend using them only on the bore diameter for which they were designed, and without an added patch?