Well the two worst things I see are the welds at the front trunnion, receiver to the trunnion and the trunnion to the barrel.
The receiver welds look like they're welded right to the rivet stubs left in the trunnion and the little pieces of the rivets are breaking loose.
The barrel weld to the trunnion is the nasty looking one, it does look like you might be able to cut enough away to get it back out, but your going to need to be very careful if you try this.
This is how I'd attack it
First drill those welds on the receiver out just enough to break the surface of the receiver so you can get the trunnion out.
Then coming from the bottom of the trunnion I'd start cutting on the weld that's on the barrel. Get it as close to the barrel as possible, then clean up the rest of the welds once they are seperated.