Take it to your local repare tech. Last time I had this problem he didn't even charge me but it shouldn't be more that $20. He will either sand down the cork or just replace it with some that is thinner.
Welcome to SOTW. You can sand down that neck cork yourself and save a trip - it is easy to do. Cut a thin strip of sandpaper, wrap some tape around the neck below the cork so you don't mar the neck's finish (or if you are careful, it is easy to avoid hitting the brass so you won't need to tape off the area below the cork - your choice) and sand around the cork by pulling the sandpaper strip back and forth around the full circumference of the cork until the new mouthpiece slides on, using cork grease. Be careful not to remove too much or you will most likely need to make that trip.
Mouthpieces vary in barrel diameter - nothing wrong with anything; it is just the way mouthpieces and necks and corks are made. DAVE
Thank you all for the warm welcome and advice! It is very greatly appreciated.What Dave said. If you haven't tried cork grease, do that first. It might be all that's needed.
I think I will try the sandpaper route (since I have a ton of it laying around anyway). I did try some cork grease, but still couldn't get the mpc to go on far enough. I will post back (probably later today) with the results.