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You might be interested in this paper, which describes morse theory on a certain class of acyclic poset-enriched categories via localization: arxiv.org/abs/1510.01907 If this is the sort of thing you are looking for, let me know and I can describe it in an answer.
Every time I read this answer, I learn something more. Thank you for taking the time to set everything down so carefully, and moreover, for answering the question that I should have asked!
Note that sets like $\{|f(x)| < p\}$ are equivalent to $\{|f(x)|^2 < p^2\}$ and the norm-squared function is much friendlier to work with (eg it is smooth). I don't know if your desired result has appeared anywhere, but I think a good starting point is Durfee's paper (jstor.org/stable/1999065). This does what you want globally, ie., without the $\{|x|<R\}$ part; to make the equivalence local, you have to show that the gradient vector field of $-|f|^2$ points inwards along the boundary $\{|x|=R\} - \{f=0\}$.
Thank you for this answer (and your book, from which I learned Morse theory!) One question: I don't see how you get $$f(x^1,\ldots,x^n) = \text{stuff}$$ when $f$ is only defined on $X$ consisting of the first $m$ coordinates. Should that $n$ be an $m$, or are you using an extension? And similarly, I'm not sure what the $b$ is in the index set of your box.
The space $B-A$ excludes the equator, and in general I'd expect the limiting gradient line to actually lie on the tangent at the equator (even in a hyperplane with parallel downward flow). This is fine as long as off the equator we don't have any points of tangency.
