You can definitely achieve this with "Envelope Deformation". I think you were almost there, but ignored a slight rotation in the original. If you rotate the original by roughly 15° (clockwise), then model your envelope and finally rotate your object back, you should be fine.
Here's what I got:
Steps to follow in Inkscape:
Pick a good font. I think for the final look this is more important than anything else. I chose BadaBoom BB. Add the desired text (e.g. "GROAR") in normal orientation.
Style the text (black 5px stroke, vertical three-point gradient spanning the entire line height).
Do the envelope deformation. At this point you have to convert the text to a path, which is a bit of a pity, as you lose editability. Bend the top and bottom paths like so:
Note: You can play with the spline nodes in each path to increase the "turning effect" of the letters. Sometimes, the outline of the deformed object may react in "funky ways" (like you mentioned), however, with a bit of node tweaking and moving you should be able to find a constellation in which everything looks proper.
Rotate your object counter-clockwise by 15°.
(Optional) Add the comic shadow effect from the original.
A final suggestion: If you want to produce a lot of text objects with this effect (e.g. to illustrate a cartoon), you may want to use "Put on Path" (from the "Text" menu) instead of "Envelope Deformation". It will produce the same turning effect (no scaling/deformation, though) and keep your text editable.