What I can imagine from your reply is as follow:

Considering Stopper/Guide force at a support.

The case F2(frictionless) consists of the thermal force "T1" totally coming from the thermal expansion of the system. No the influence of friction


The case F2(friction) consists of thermal force "T2" and friction force "FF". The "T2" is not the same as the "T1" because it has friction influence which makes the thermal expansion of the system in this case different from the case of frictionless

That's why F2(friction)-F2(frictionless) doesn't give the correct FF value

Correct me if any misunderstanding