Scale size of \nmid

[sleek arrow]

There are some cases I need to use \bigm| instead of \mid for legibility (primarily when combining with absolute value bars; e.g., Lagrange's Theorem H \le G \implies \lvert H\rvert \bigm| \lvert G\rvert). However, I'm not sure how to generalize this to \nmid. \not\bigm| looks terrible (|/), as expected, and \bigm\nmid doesn't work at all since \nmid is not a delimiter.
Is there some way to do this that looks good? Maybe some sort of "small \not" that looks right when used with \bigm|.

[thorny steeple]

<centernot> $\centernot{\biggm\vert}$

[thorny steeple]

,ctan centernot

[slender cobalt]

centernot – Centred \not command

The package provides \centernot that prints the symbol \not on the following argument. Unlike the default \not command, the symbol is horizontally centered. The package is part of the oberdiek bundle.
​ ​ ​ ​ ​ ​ Sources: /macros/latex/contrib/oberdiek/centernot.dtx
Documentation: Package documentation
​ ​ ​ ​ ​ ​ Version: 1.4 2016-05-16
​ Contained in: TeX Live as oberdiek, MiKTeX as oberdiek
​ ​ ​ ​ ​ ​ ​ Topics: Maths
Read more at CTAN page of the package.

[sleek arrow]

interesting, but the slash is still way too large and looks weird in practice

[thorny steeple]

$\ooalign{\hfil$\biggm\vert$\hfil\cr\hfil$\diagup$\hfil}$

[thorny steeple]

$\ooalign{\hfil$\biggm\vert$\hfil\cr\hfil\textfractionsolidus\hfil}$