How to compute gamma for at-the-money regular calls and puts when they approach expiration to avoid explosion of portfolio's gamma?

When and at-the-money regular call or put approaches expiration, gamma tends to infinity. However, for practical purposes, there is only a finite change in delta. The problem is that if any of the options in your portfolio gets a crazy high number, this ruins the usefulness of the gamma of the whole portfolio.

I guess that that for practical applications you can choose a fixed change dx in the value of the underlying and compute gamma numerically using the pricing function f(x):

= ( (f(x+dx)-f(x)/dx - (f(x)-f(x-dx)/dx )/dx
= ( f(x+dx) - 2*(f(x) - f(x-dx)) / dx^2

The absolute value of this numerically computed gamma could be used as a ceiling for the absolute value of the closed-form formula-computed gamma.

Ideally, dx could be something like the standard deviation of daily price changes, or any number (of this order of magnitude) that the traders find easy to mentally manage.

I wrote the above just looking at the crystal ball. How do traders and financial engineers tackle this problem in practice?

There is a greek for $$\frac{\partial \Gamma}{\partial S}$$, it's called speed.