GR defines velocities in non-expanding space. But the problem with expanding space is, that we do not have in GR a definition of velocity in expanding space.
What you are referring to, is that two objects (galaxies) are receding from each other because space itself is expanding between them. Though, the galaxies might actually even be moving locally towards each other through space, they are still receding from each other because space itself is expanding between them. This is because these galaxies might actually be moving in their own galaxy cluster towards each other, that is, their speed vectors show a direction towards each other. If you would locally measure the speed of one galaxy (A) compared to another (third) galaxy close to it, you could see that the galaxy is moving in the direction of the other (B) galaxy. This is if you view it from the local frame. So you would see that A is moving towards B when viewed from A's frame. But when you look at A and B from outside the whole galaxy cluster (that involved both A and B), then you would see them receding. You would see the whole galaxy cluster expanding. Every galaxy in the cluster is receding from the other one (in this example). This happens because the whole space that the cluster is in, is expanding.
You are right that when an object moves through space, as it accelerates, it gains more energy (momentum). But since in your case, the objects are not moving through space faster and faster, they do not gain more momentum. It is space itself that expands, causing them to be further from each other. There is no acceleration through space, because in their local (in the galaxy itself) frame they are not accelerating. As per GR, they are not accelerating.
Space between them is expanding, and the expansion is accelerating, but as per GR, this is not the definition of acceleration of an object. Because of this, they will not gain energy, and their stress-energy will not become bigger. They will not become black holes.