I want to be able to start with the raw white text (font is bank gothic) and replicate the shadow effect pixel for pixel (or as closely as possible).
I have tried many different blur effects in GIMP but just can't get it looking exactly the same.
Something quite similar (not "precisely") can be done semi-automatically with the Layer Effects plug-in in GIMP, and a little extra manual work.
To install the plugin save the layerfx.2.8.py file in your GIMP plug-ins folder, and relaunch the application.
With a text layer selected, click Layers > Layer Effects > Drop Shadow
Use these settings shown below. You may need to adjust the size and distance setting depending on the size of the text.
Then you could select the green shadow layer, apply a transform (Shift+T) to stretch it horizontally, and squish it vertically a bit.
This is the result
It's also possible to duplicate the green shadow layer, and adjust the position and opacities, and get something a bit more random