You are not going to get it out without removing the front housing.

Just buck up and do it right, the welder idea has me thinking about all
the embers and splatter that will ruin your gears from grit.

The only other thing I can think of would be to pull the drivers side, get a
long thin rod and poke it through to the broken piece and hit it with a hammer
to knock it out, otherwise pull the front

Wild

and if the long rod doesent work, your not very far from pulling the diff anyway