I don’t have time to carefully look at the linear algebra but the basic idea is correct. See also The QR Decomposition For Regression Models for a demonstration of a similar problem, including the translation of a prior density in the nominal parameterization to the reduced one.
Ultimately it helps to recognize that you’re reparameterizing the entire model, not just the one likelihood function, so you need to propagate the change everywhere including through the prior density with the requisite Jacobian and what not.
The reduced form works out parameters that are more directly informed by the available data that should facilitate the fit, at least provided that the transformed prior model isn’t awkward. Otherwise the main options are a more informative prior model – even more information on just a few parameters can help fight the degeneracy – or incorporating complementary measurements that identify how some of the reactions work on their own.