After some reflection, I think I came to an understanding of what should be the answer to the problem. Transformations \eqref{eqB1} and \eqref{eqB2} should be viewed not only as transformations between coordinates but also as perturbations around the original coordinates such that at the boundary where $\varepsilon\rightarrow 0$ these must be retrieved to recover. In this sense, in order for such a criterion to be satisfied, we must choose (or configure) the functions $\mathcal{T}$ and $\mathcal{Q}^\mu$ as $\mathcal{T}\big(t,q^\mu\big) = t$ and $\mathcal{Q}^\mu = q^\mu$.

Basically, I think this is the same answer that @fredwhileshavin wanted to give in the previous post, but in a veiled way.

Well, I think I'm on the right track for a good answer, but I leave it to the community for judgment.

After some reflection, I think I came to an understanding of what should be the answer to the problem. Transformations \eqref{eqB1} and \eqref{eqB2} should be viewed not only as transformations between coordinates but also as perturbations around the original coordinates such that at the boundary where $\varepsilon\rightarrow 0$ these must be retrieved to recover. In this sense, in order for such a criterion to be satisfied, we must choose (or configure) the functions $\mathcal{T}$ and $\mathcal{Q}^\mu$ as $\mathcal{T}\big(t,q^\mu\big) = t$ and $\mathcal{Q}^\mu\big(t,q^\mu\big) = q^\mu$.

Basically, I think this is the same answer that @fredwhileshavin wanted to give in the previous post, but in a veiled way.

Well, I think I'm on the right track for a good answer, but I leave it to the community for judgment.