Simply never use a different gun than you start with. It really isn't as daunting a task as it sounds considering that the plasma cutter is actually the most versatile weapon in the game. You can own the others just as long as you don't use them. Obviously, you'll want to max out the stats of the Plasma Cutter as soon as possible. You only need to worry about the stat increase nodes to max out a weapon. Only fill an empty slot if you need to in order to reach another stat slot.
If you wanted, you could use the New Game+ feature to load an already beaten save file to do this run through with an already maxed out weapon. This is completely unnecessary though as you should have the weapon maxed long before running into enemies that are overly difficult to beat.
If completed correctly, it should unlock just as the credits start to roll.