As you,
@Winters, may guess from my avatar, I have cleaned my Long Land Pattern King's Musket (Okay, Brown Bess) more than a time or two. If all you have is the parts that came with a Pedersoli Bess and the wherewithal to shoot it, that implies that you only have the steel ram rod that came with it. The threads are a metric thread that won't match up with parts available at most local gun shops. You are correct that the bulbous tip isn't too likely to grab a cleaning patch. You will need bamboo skewer which is bigger than a toothpick to plug the touch hole. You will need some sort of soft cloth to use as a cleaning rag. You may have to sacrifice a t-shirt to get cleaning patch material. Yo should have a string to tie the patch to the bulbous tip on the ramrod a leave an end long enough to be able to pull the patch from the barrel if necessary. Cold water, warm water, hot water or boiling water with a bit of dish detergent soap will effectively clean the fouling from the bore of your smoothbore (its not a rifle) Brown Bess. Soap and water is also used for cleaning the lock. Use as many patches as you need to clean the bore. A grayish patch is good enough. Use dry patches to remove as much of the water as possible. Dampen a patch with rubbing Since you don't have a good rust inhibiting lubricant, you can use any of the household cooking oils, Canola, corn, olive, whatever, to protect the metal while you get a can of Birchwood Casey Barricade.
With something unimportant you can cut a groove or cut the end to form a slit lengthwise. A piece of string can be used to tie a cloth patch to the end.
If you have no mop but can warm the gun near a stove don't clean it until you can properly.
If you have access to a hardware store a 1/2" of 5/8" dowel rod 48" long can be used as Britsmoothy suggests. Use a kitchen paring knife to carve a few circumferential grooves around one end of the dowel rod to secure the patch. Within muzzle loading, you have lots of options to get your new Bess clean.
Yes, patina in the form of the spots will always appear. Steel wool, the gray Scotchbrite pads (which you can buy at the hardware store when you buy the dowel rod) will clean most of those spots off.
Now that you have a patch stuck in the bore, try turning your rod clockwise to break it loose now that the patch is damp. You will need something to grip the end of your rod securely to pull it out. May take a tug or two to get things moving. I have a bench vise to capture the end of the rod. On most of my rods I can thread a cross Tee to gain advantage to pull it out. I have also used a strip if leather boot lace to tie a timber hitch on the exposed rod to tie to a secure post to pull the rod and patch out.