I will just say that this sounds like a question that is very hard to answer in a general way, because the possible effects of the new physics are so diverse.
One specific scenario that comes to mind, is dark matter that sometimes impacts atomic nuclei. This might occasionally jog an atom in the atmosphere, creating a noise term in atmospheric dynamics, and you could try to quantify its effects.
For example, there must be models of atmospheric hydrodynamics in which there are phenomenological parameters that in principle could be deduced solely by stipulating atmospheric composition, Earth's size and distance from the sun, and so on, and then using physical first principles to determine e.g. the thermodynamic and hydrodynamic effects of the sun's radiation on such an atmosphere.
If the noise term due to nuclear recoil from dark matter impacts actually modified those phenomenological parameters - e.g. by persistently introducing energy into a vibrational mode of a specific molecule, in excess of what first-principles calculations predict - that would presumably be an example of macroscopic deviation from standard model expectations.
I originally thought of this scenario because the question mentioned chaos, and I tried to think of some dynamical system where a difference in fundamental physics might be chaotically amplified.
But even in this case, if you were testing for this effect, you wouldn't need to somehow simulate the whole atmosphere as it was, starting at GMT midnight 1 January 2015, and then see if the weather turned out differently from reality; because the effect shows up, first of all, in a deviation from theoretical expectations, in the energetic behavior of some molecule.
The lesson I draw from this, is that even everyday life, despite its complexity, is made up of specific classes of physical system; and that if you are looking for a subtle impact on everyday life, of hypothetical new fundamental physics, it won't come about in a mysteriously generic way because of little perturbations distributed across numerous heterogeneous events. The perturbations will arise in a highly specific way, because the new physics impacts certain highly specific physical systems, and it should therefore show up in focused physical studies.