How can we explain that Temperature is a classically frame-independent quantity to high school kids?
If you stick to gases then things are relatively straightforward because the temperature is related to the relative velocity of the gas molecules, that is the velocity of the gas molecules relative to each other.
If you put your canister of gas in a fast moving (but non-relativistic) rocket moving at some velocity $v$ then you add the same velocity $v$ to the velocity of every gas molecule. But when you calculate the relative velocities of the gas molecules the extra velocity $v$ just cancels out and the relative velocities are unchanged. That means the temperature is unchanged as well.
There was a related discussion in Why isn't water running faster hotter? that you might also want to look at.
I specified a gas because we have a nice clear relationship between velocity and temperature via the Maxwell-Boltzmann distribution. I don't know how deep you want to go with your class, maybe going into the details of how this distribution is derived would be a bit much.
We can do that using states of Matter. If temperature is frame dependent, the observers in different frames should observe different states of matter near Melting and Boiling points which is not the case. This was the easiest explanation I could think of.