I'm trying to think about what could cause this. There's one problem for sure - the Snowrisu should be dead forever. Did you also kill the Snowrocco? (A second one should show up.) After they're both dead, a second cutscene should occur - the camera should scroll to the right, open the gate, then bring the camera back. Also after that, the character should be able to walk through the gate and the rest of the level should be within camera view now.
I just tested this on firefox on itch.io, and the snowman fight works as intended in this browser setup. I did notice a bug where the "powerup" at the end was not working as expected - it's supposed to move around as a demonstration of how it works, but the powerup is motionless. I didn't see any errors to explain that one, I'll have to keep looking.