itisus, your question is a good one and I don't know the answer. It may well be that nobody does exactly but I expect there are a lot of theories. It is not certain what is meant by mass itself (gravitational or inertial) or how much the mass of a body we observe is actually the energy of the gravitational scalar field. I think the easiest way to think about this is to consider only a two body problem of an object and a non-rotating black hole. An inert observer being permitted.

Let's suppose the observer is positioned a long way off but perpendicular to a line drawn between the BH and the object. Let's also suppose the BH and object somehow start off stationary with respect to each other and the observer. There will be a potential energy between the object and the BH and, as the two are drawn to each other this will translate into kinetic energy. Both object and, to a much lesser extent, the BH will accelerate towards each other. Eventually the object will disappear into the EH of the BH and, by conservation of momentum, the resulting object will again be stationary. The total energy of the system will be the same, so the BH will grow, as viewed by the observer, according to the energy gained by the impact of the object. This will be the sum of it's stationary mass and of the mass due to the velocity at impact of the object. The energy gain should be the same as calculated from the initial potential energy.

Whether it is fair to make the simplifications here is open to conjecture. Some theories, probably the most accepted ones, have mass as only having meaning with respect to the total mass in the universe, so extrapolating to a two body system, and one that ends up as one body, as in this case, may not be valid. Also the position of the observer should not matter, but I'm not wholly convinced that this is the case here; I think it's OK but the situation can get very complex.

The point you make about getting an infinite energy situation seems a tricky problem. If the object was positioned initially a similar distance from the BH as the observer, then a second observer on the object would also perceive the BH as having an EH. The consequence of this is that when he falls through it he will, as seen by the first observer, to have infinite energy. However this seems in contradiction to the finite potential energy at which the two objects start off.

I think I may be missing something obvious. Maybe someone on the NS team knows, or at least knows someone who does???