In some ways this is a very deep question, and in some ways, it seems like you are a bit muddled about what an "observable" is.
Certainly, one could imagine a BSM theory in which particles have new kinds of properties not found in the SM. Indeed, as often as not, they do. But, to have a new observable, you would need a new way to make observations than the ones currently available to us.
For example, many SUSY theories have a property called "R" which is conserved or not imperfectly conserved. But, R isn't an "observable" in this theory. One infers the "R" number of particles by figuring out what decays of other particles happen and what decays never happen, even though they would but for R conservation.
Indeed, there are important properties of particles in the SM that are not observables, such as QCD color charge, which is inferred, but can't be directly measured. Even the non-top quark masses are not themselves observables in the SM, due to the confinement of all quarks in hadrons except the top quark which doesn't hadronize, even though they are fundamental parameters of the theory. But, if one had a BSM theory in which the gluon fields of SM QCD could be suppressed somehow allowing for free quarks, the up, down, strange, charm, and bottom quark masses might become observables in the BSM theory.
This is another ambiguity in your question. Certainly, a BSM theory could have new quantities that can be observed. For example, in SM4 model with four generations of fermions instead of three, there would be a tau prime lepton which would have an observable mass, spin, parity, charge, half life, etc.
But, it seems to me that this is not what you are really asking. Instead, you are asking if a BSM theory could have a new "kind" of observable, rather than just a new phenomena that can be experimentally observed (which almost by definition would be true of almost every BSM theory except a true "within the Standard Model" BSM theory that merely explains the properties of the SM at a deep level without actually proposing any phenomenological differences at all, and maybe that's a WSM theory for "within the Standard Model" rather than a BSM theory).
In principle, a BSM theory could include both a new property of particles and an associated way to directly observe that property.
It could even be something that scientists have seen forever but never previously thought to conceptualize as a separate meaningful fundamental "observable" like "chirality" which we usually think about in everyday life as merely a non-fundamental emergent property of the location of different particles in a larger system at the same time rather than as a property that a point-like property could have.
Something along the same lines in string theory is the character of a string as an open string or closed string in a particle that is fundamental although not truly point-like. I could imagine that there could be a way to directly observe the character of a particle as open-string-like or closed-string-like thereby creating a new observable that really doesn't exactly correspond to any other existing observables even though, like a point-particle's chirality, it has a classical/macroscopic physics analogy which is emergent rather than fundamental.
For example, by analogy to your suggested "Kaluza observable" one can imagine that there might be an observable property of space-time such that the apparent speed of light though that area of space-time which appears to be a vacuum is <c because it is actually riddled with unavoidable detours through an extra Kaluza-Klein dimension, unlike the ordinary parts of space-time that we know and live.
Even there, it is hard to say if the property of space-time we are observing is really an "observable" as we aren't really measuring something different that time discrepancies and distance measurements and speed and momentum - the stuff of Newtonian mechanics as modified conceptually so that it can have rigor in the context of general relativity and special relativity. Deep down, we are really just observing when and where photons are launched and when and where they land, and only indirectly observing the different property that the space-time between the launching spot and the landing spot seems to exhibit.
Similarly, the concept of a negative probability is mathematically well defined and doesn't generate impossible events so long as it infers with another set of positive probabilities, so that the "observables" (i.e. how often something happens when all the relevant probabilities are added up) is always positive and real. Negative probabilities, imaginary numbers and all sorts of other properties of the world that don't exist in the SM that don't create non-physical true observables are all good and well and lots of BSM theories propose new ones, but they aren't observables themselves.
One could imagine a BSM theory in which there is "negative" mass, but it only appears in a manner that interferes with "positive" mass terms, resulting in net positive mass.
Part of the problem in defining "observable" is figuring out how to draw the line between direct and indirect observation in a way that meaningfully distinguishes a non-observable property that only explains the results with unobservable intermediate steps, and a direct observable, when in real life with all of its true complexity, the chain of events from a physical event that happens to the layers of hardware, software and wetware that intervenes between the physical event and our awareness of that physical event as human beings is so multi-layered and complex.
