If the mouthpiece is that stuck, it's probably gunked up and corroded inside. The other posts are good ideas, but also try these methods.

Get some penetrating oil (Corrosion Cracker brand is great, but anything at the local hardware store is good). Put a few drops where the mouthpiece enters the leadpipe. If you have a torch, heat up the leadpipe a bit to help draw the oil in. Let the oil work overnight and try again tomorrow.

Put the bobcat on, turn the screws to apply pressure. Leave overnight and see if it's out tomorrow morning.

Once you get it out, clean the mouthpiece shank really well. Rubbing alcohol can help. Steel wool is the next step up. Clean inside the leadpipe as best you can.