You aren't talking about adding a new monster, you are talking about changing the game completely. The 'monsters' in the game right now are simply control points. They are basically the same as dominion towers except they attack back. They exist as strategic points on the map that encourage confrontation. If they didn't exist the game would potentially be an extremely boring lane only game.
The monsters in the game don't move for a reason - a random, aggressive, moving strategic control point is no longer a strategic control point but a freaking monster that may or may not be in the game at any specific point.
Quit thinking of the monsters in the game right now as monsters and realize they are simply control points. You'll quickly see how your idea is not a simple 'add a new monster' to the game idea but really a 'change the game to add in monsters' idea.