In case the original authors ever come across this article: There is a standard for writing Mathematica modules! Please take a look at some other modules available online, and see how they split the code into small, reusable functions.
Even the name is too short. In the Mathematica naming convention it should be called "InverseLaplaceTransformCME[...]" or something like that. Ideally, use the same calling convention as the built-in function, documented here: https://reference.wolfram.com/language/ref/InverseLaplaceTra...
This would allow your function to be a drop-in replacement, allowing users to switch between the symbolic and approximate versions trivially.
You may even want to contact Wolfram Research! They just implemented a new "Asymptotics" module that includes approximate inverse Laplace transforms as a feature. See: https://reference.wolfram.com/language/guide/Asymptotics.htm...
They might add your approach into the 12.2 release, which would mean that many thousands of people could automatically benefit from your hard work!