Will assuming that none of the charges are fixed, change the potential energy of the two point charge system?

So first of all realize that you are dealing with a definition within Electrostatics, which means there is no time involved, no dynamics, everything is to be thought as frozen in time. So the question, "if we assume none of the charges to be fixed" makes no sense in this context because we are looking at one snapshot in time and calculating if you want the instantaneous potential energy.

Having said that, this idea of bringing in charges from infinity to a given point is a way to understand and compute the potential energy of a configuration, however it does not correspond to any real movement of charges.

Consider the system of two point-charges you mention. You also have to bring the first charge to the desired point from infinity, however since there is no other charge already present, you are not doing work against anything, so the first one is "free". Then to place the second one you have to some work as you computed it. And no matter how you do this since the forces here are conservative, you will obtain the same answer. Again there is no time here.

Now if we ask ourselves whether the situation is stable, the answer is obviously not. Let say both charges are positive, if you were to let the time run (unfreeze your simulation) they will fly apart. This is probably what you had in mind.

So "fixed" in electrostatics just means there are no time dependencies, charge distributions are to be thought to be frozen in time, or being examined at one instance in time. Computing the potential energy at one instant in time, can be done in many ways, but mathematically it agrees with this idea of bringing in charges from infinity.