Thanks for adding this to the forum. I will take a look into this. Your initial analysis sounds correct - it is because items are being shared from multiple devices however beorg only incorporates shared items into files when it comes into the foreground. At that time it is likely that different versions of the files are being updated because this is happening on multiple devices. I'll have to think if there is a way of mitigating this. Whilst plain text files (which beorg uses) are great they don't necessarily play well with this kind of behaviour unfortunately.
Syncing in the background on iOS is something over which the app only has limited control. Whilst the app can request that a background refresh happens it is iOS that ultimately controls how often that actually takes place. So a background sync happening within 15 minutes is unfortunately optimistic - it might be a few times a day.
I will look into how this can be improved but a fix is not likely to be immediate. You might be better creating a Shortcut and using that to share to beorg. Take a look at https://beorg.app/learning/shortcuts-web-pages/ for an example of this.