Consider the situation from the point of view of an observer who is falling into the black hole along with the mass. The observer can be a little ahead, a little behind, or off to one side. The observer does not see the object accelerate to any spectacular velocity. If the observer is ahead of the object, then they see the object start from rest and then accelerate outward a little, relative to them. As the observer and the object pass the event horizon, nothing special happens to them. The observer's clock keeps ticking for the remaining time before they hit the singularity, and during that remaining time they can continue to observe the object. I think you can see from this description that it wouldn't make sense to expect the energy of the object to hit infinity just because it's at the moment of passing the horizon. If it did, then what would happen after that?
General relativity doesn't have global frames of reference, so it doesn't make sense to talk about the infalling mass's velocity or energy in the frame of reference of a distant observer. Popularizations often talk this way, but it's wrong.
There actually is no measure of mass-energy that is always conserved in all relativistic spacetimes. There is such a mass-energy in the case of a Schwarzschild black hole, but only because that spacetime is asymptotically flat, i.e., it describes an isolated object. That mass-energy is a property of the whole system and stays constant at all times, as measured by an observer who is far away compared to the infalling mass. That observer can measure the black hole's mass-energy by the gravitational forces it exerts on distant objects. It doesn't go to infinity. It stays constant. This is exactly what you would observe if an asteroid crashed into the sun. As measured by an observer a light-year away, the mass-energy of our solar system would not change at all during that process.