Your equations do not display correctly. Please, review your LateX code. Give it a try on The Sandbox, eg,
Often you can run your code piecewise there and usually you can isolate the part of the code that's making it fail to compile.
Nevertheless, I see matters of basic concepts/principle etc that tell me your idea cannot possibly be right. Introducing a particular set of coordinates to tag space-time points should not affect any calculations in any sensible physical theory. When, eg, you write the Lagrangian piece,
\[ \frac{1}{2}(\partial_{\mu}\phi)_{\text{Bas}}(\partial^{\mu}\phi)_{\text{Bas}}-V_{\text{Bas}}(\phi) \]
functions such as a scalar field or a Higgs-like potential should not depend on the particular coordinate choice, on account of both being scalars. Sub-indexing "Bas" meaning "evaluated in my coordinates" should not make any difference.
So nothing you propose seems very sound as per the formalism.
I hope that's clear and was helpful.