After swapping some messages with the author of Beorg, I made a new version of my Shortcut to make use of the x-callback-URL feature which removes some steps in the process. No iOS share sheet is shown now. The setup steps I listed before still hold true.
※ there are a few odd calendar entries that do not fully populate in Beorg. I am troubleshooting and will open a trouble ticket if needed. YMMV.
Update: for those wanting to use the iOS Share Sheet version, it is still available here.