It is my understanding that physicists hope to use conservation of energy in the LHC to determine if there are multiple dimensions AND if there is dark matter. Let's suppose that through conservation of momentum, we detect a particle that apparently only reacts through gravity, and there is some amount of energy lost. Would we just E=mcÂ˛ to determine how massive the particle is? What if more energy is lost via extra dimensions?
The missing energy is just one "dimension" to the problem. In practice, you need many more (generalized) "coordinates". Different physics will show different behaviors in the interplay (correlations) between those coordinates, which has to be studied by simulation but is not necessarily model-dependent^{(*)}. In the literature, this us called "signature" for dark-matter for instance. (*) For instance, you have a model predicting the existence of a spin 1/2 particle. Given the assumptions of this model, you calculate a cross-section. The angular decay in the rest-frame of your model-dependent particle is itself model-independent (or better, depends not on the assumptions of your own model, but only on Lorentz invariance)
So with all of that other information they can determine how much of the missing energy is actually the mass of the particle?
Depending on which reaction exactly, it may or may not be done on an event-by-event basis. But given the entire sample, using statistical analysis, yes. You may want to read about "Dalitz plots" for instance. Note that, otherwise (if they could not determine the mass of the missing particle), they would simply not understand anything !
Also, note a very general thing : IF you manage to find an example where even in principle you can NOT distinguish between the two final states (say missing particle VS extra-dimension), THEN the two processes interfere (you can not add cross-sections, you must add the amplitudes). Typically, the production of your particle would be enhanced in some kinematical regions and suppressed somewhere else, due to extra-dimensions. Now in this case, it's not so trivial to make general statements. But you can already imagine that the dependencies become more complicated, say richer, and it's not impossible that you get a better handle on your parameters. But I expect that proponents of extra-dimensions would in any case like to cross check the results using another channel without this specific particle.