No, it is not possible to differentiate between these two interpretations because they're ultimately physically equivalent.
First, we should separate the discussions in cases where the total energy is conserved and where it isn't. The existence of conserved energy in general relativity (which must be ADM-like) actually requires vanishing or negative cosmological constant – the spacetime is Minkowski or AdS. In de Sitter space, there's no nonzero gauge-invariant definition of energy that would be conserved because the de Sitter space has no asymptotic region at infinity.
In the de Sitter space, masses of objects may therefore change in various general ways and by measuring them, you can't deduce pretty much anything.
In Minkowski or AdS space, the energy is conserved. Let's consider an anti de Sitter space with a negative cosmological constant. This means $\rho\lt 0$, a negative energy density, with a positive pressure $p=-\rho\gt 0$. The energy of the mass that ends up as the black hole is conserved, it's the total energy in the spacetime, assuming that everything collapses. However, the value of this total mass/energy is given before the black hole is formed – it stays the same by the conservation law – which means that we can't deduce anything new if we measure the same value at the end.
What you really want to do is to "attribute" or "divide" the total mass/energy of the black hole into different regions – either the generic black hole interior or the singularity. But this "attribution" or "localization" of matter is exactly what is impossible according to general relativity. The conserved total mass/energy cannot be written as an integral of a well-defined energy density. Such a thing may only be written in the "Newtonian" limit of weak gravitational fields and the existence of black hole is exactly the opposite situation in which the "weak fields" condition is dramatically violated.
So no, your verbal descriptions of the situations are just heuristic and to see what actually happens, you need to discuss things quantitatively, using the right concepts suggested by general relativity and using the right equations. The (Einstein's) equations say a very clear thing about the impact of cosmological constant in the absence of black holes much like in their presence and any idea about "two possibilities" (the cosmological constant is a property of space or a form of energy) is a mere illusion, an artifact of non-quantitative thinking about the problem.