Well, I finally got to the bottom of the issue. It seems that in order to use Offline mode, the user must have admin rights on the computer. This may be mentioned somewhere in the documentation, but there is certainly no mention of this on the options page where you actually check the box. I feel like this should be advertised more prominently. Often times users in a corporate setting are not granted admin rights, and I can see this happening many times over to unsuspecting ArcPro users. ESRI support was very helpful, though (big shout out to Bindeshwari!). The crashing stopped after I renamed the hidden ESRI_Licensing folder in the local appdata folder, but in order to access my offline organizational account, the account had to be reset, which took a few days. In the meantime, I was able to create a trial account and use that.
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