I am trying to re-create Instagram's structure effect within Lightroom, but I can’t quite figure it out. Clarity, Sharpening, and Dehaze were my best bets but things are still very different.
Is this possible?
I don't think that it's entirely possible. The effect looks to be a heavy tone mapping, which Lightroom can't do exactly. In Lightroom, Clarity, Contrast, and Tone Curve may get you quite a bit of the way there. Also, you may need to add a brush over the entire image and increase those settings on that as well to duplicate/multiply the effect.
Edited to add: You may also want to Google "dragan effect lightroom". It's a method that may produce the effect you want, there are free presets available.