We know from conservation of angular momentum that $I$$\omega$ = constant. So when the object is heated, the body expands leading to a change in moment of inertia. If the new M.O.I. is $I$2; then as $I$2 > $I$1 , automatically $\omega$1 < $\omega$2. Therefore the problem of decreasing rotational kinetic energy is solved, as work is done to expand the material.
Note: Expansion takes place partly from the heat supplied too. As values are not given we cannot calculate for sure.
The principle of conservation of angular momentum says that angular momentum remains conserved unless an external torque acts on it. The net torque on a body is defined as:
$$\vec{\tau\,}=\dfrac{\mathrm d\vec{L\,}}{\mathrm dt}$$
We can clearly see from this definition that since external torque on the body is zero, the angular momentum is going to remain constant. But the angular velocity is not, and that is what which changes with change in angular momentum, because:
$$\vec{L\,} = I\vec{\omega\,}$$
For example, ice skaters when have their arms outstretched, their moment of inertia is high and so angular velocity is low, but if they draw in their arms, their moment of inertia decreases and correspondingly, without any external torque, their angular speed increases!
Edit:
The revision to your question has made it further interesting. Imagine that the rod is connected to a motor. Now, once the insect starts crawling towards the end, the moment of inertia of the entire system increases. According to our equations, the angular speed of the system should correspondingly decrease. But, we are told that it remains a constant $\omega$. This means that the motor has continuously apply a torque to keep the angular velocity constant! This is the external torque that we have find in the question, and it is responsible for the increasing angular momentum as well as kinetic energy of the system.
In the first case, there was no constraint keeping the angular velocity constant, unlike your second question.
Best Answer
As pointed out in many comments, the concept of Work is integral here.
In Newtonian Mechanics, we have the equation: $$K_{initial} + W = K_{final}$$ Here the work is done by changing the $I$, the moment of inertia. To do this, the mass distribution of the body needs to be shifted further from the axis of rotation.
Naturally, at least some particles of the body need to be moved away from the axis of rotation. To do this an external force is required. Displacement of these particle is against the centripetal force causing the rotation - such as tension. This leads to the particle, and hence the system, losing Kinetic Energy. (Negative work is being done by the centripetal force on he particle)
Exactly how much energy is lost is given by the $\Delta E$ you have caculated.
The above is just an attempt to provide an intuitive view of conservation of angular momentum in view of linear dynamics.