Really fun and interesting idea! Fits the them perfectly :)
I was able to "cheat" sections, by standing on objects and moving the character to high ledges and over the lava. Maybe look to detect when the play is doing this (Raycast down etc) and cancel the mouse's hold on the object.
I also only took one box through instead of 3 and used it to push the player through the barriers. I think a system to cancel the mouse's control over the object when it interacts with the player would solve most issues like this.
Overall I really think the idea has a lot of potential. Amazing work!