I've seen it said before that we often ignore potential energy in relativity because it can be included in the mass term. It is commonly said that a hydrogen atom has less mass than the sum of its parts, due to the negative binding potential energy, for example. But I haven't seen an intuitive argument for why this would be the case. In fact, it seems counterintuitive.

Suppose you have two equal masses, separated by a distance $d$. If they are both stationary, and noninteracting, then their total mass is $2m$. If you were to apply some force $F$ for an instant $dt$ to the first mass, and the same force to the second mass, the system's center of mass would gain a velocity $dv=Fdt/m$. While center of mass is not talked about often in relativity, I understand that it does have a definition, though obscure.

Now suppose one of these masses has a positive charge, and the other a negative charge, of equal magnitudes. There is now a potential energy of interaction which differs from the potential energy the masses would have if they were infinitely far apart. While it is true that they will attract each other, their changes in motion oppose each other. Applying the same force to both charges, the center of mass seems that it should still gain the same $dv$; in fact, the deviations in the two individual masses from this velocity due to their mutual attraction should exactly cancel. It seems that mass 1 should have a velocity $dv+dv_{0}$, while mass 2 should have a velocity $dv-dv_{0}$, where $dv_{0}$ is the change due to their attraction.

So if the center of mass gains the same change in velocity for the same force in both situations, how does the system with the electrostatic potential have less mass? Is mass not related to how hard something is to accelerate? Where is my misunderstanding?