You said the missile had some amazing ability to turn, and now you're saying it has no fuel. You can't have it both ways. There is fuel, or there is no energy to spare. Being "cold" isn't generally possible; nothing cools instantly. Modern IR sensors do not rely on engine heat. Supersonic flight generates plenty of heat on the nose.
That last bit deserves more emphasis. Heat was a serious problem for the Concorde and SR-71, despite operating at much slower speeds. The missile is hypersonic. That is the speed at which air molecules come apart from the heat. IR detection will be trivial.
There is a circular circuit board that is very much needed for the proximity fuse. That is full width.
The speed of the missile contributes to its own destruction. Bullet impact will be devastating. The missile can not endure supersonic flight with a lopsided hole and a bunch of cracks, and it wouldn't fly straight even if it could. Tumbling means instant destruction, like Space Shuttle Columbia.
Shooting back is the action to maximize chance of survival. It takes priority; all other options are obsolete.