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I tried the technique using "Text -> Object To Path -> Path Effects -> Envelope Distortion", but moving edges or points distorts the image in a way incompatible with the example I posted.
In the end I found scaling a shape along the "y-axis" and using it as a guide to reposition individual nodes in the text path worked to create the effect I was looking for, but it was time consuming and repetitive. This process itself it very reproducible and could turned into an algorithm in my opinion. This leads to my original question, is there automated way to recreate this effect in Inkscape?