One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range

One way to deal with these kinds of issues is to express your initial conditions as intervals (or even distributions) in the sense that you include the a range of possible values, rather than the most probable one (which is normally implicitly done). So if you measure the earth to be (6±1)e24 kg, then you work with something that looks like [5,7]e24 kg i.e. the segment of the number line corresponding to the possible physical realities that led to your measurement. You'll get a range of different outcomes in the end, and their relative probabilities given your priors. You can do this exactly for some systems, but usually you'll do some monte carlo and hope it's valid. This is similar to classical (linear) error propagation where you carry around an "uncertainty", but chaotic systems don't generally allow you to make the assumption of narrow Gaussians used there.