How do energy changes in reactions avoid violating the conservation of momentum?
How do you measure temperature? The thermometer has to be in thermal equilibrium, so you need to send the thermometer with the object. But that means that temperature, for our purposes, is going to be invariant between different reference frames. No matter how you're moving, the reading on the thermometer is the same.

This is clearly not the case with translational kinetic energy of the bulk material. In the rest frame of the object moving with constant velocity, the kinetic energy is zero.

Therefore, you should not mix up the thermal kinetic energy of the system with the translational kinetic energy of the system. They can't be the same thing. In other words, when the exothermic system loses energy, you don't get to subtract that from the translational kinetic energy.

See also these discussions: here and here.

I would also like to point out that you've just described a rocket. And rocket science, while complicated, is still obeying conservation of energy and momentum.