As the rules currently stand, you should be able to put an immobile robot in the arena and get one point for doing so. However, then if the robot is immobile, the 10 second countdown should begin immediately.

I completely understand your situation but a key part of our hobby is not only what happens in the arena but repairing damage between rounds. There is nothing in the rules to state that it is OK to be immobile as long as that was from the last fight.