Extending Dimensional Analysis?

Question

Is there some way to extend Mathematica's dimensional analysis? For example, I'd like to be able to provide it with dimensional equations like energy=mass with units scalings like c^2 and have it extend the possible interpretations in terms of target dimensions and/or units.

Although my question is more general, the aforementioned case is supported by DimensionalCombinations if you provide the additional parameter IncludeQuantities -> "PhysicalConstants" with this example:

DimensionalCombinations[
       { QuantityVariable["E", "Energy" ]},
       QuantityVariable["m", "Mass"],
     IncludeQuantities -> "PhysicalConstants",
     GeneratedParameters -> None]

which produces these possible interpretations:

PS: I'm aware of this answer providing a kludge to extend units conversions but it isn't clear this will also extend the function of, for example, DimensionalCombinations and, in any event, I would hope for something that isn't a kludge. (And in any event, the code in that answer doesn't work on my cloud notebook copy. It produces an error upon executing the test.)