So it sounds like you really want is the https://wordpress.org/plugins/syndication-links/ which will give you a field under your content to show where the content was syndicated to (and properly mark the links up with rel-me).

I don't think brid.gy automatically puts the syndicated link into the field, so you'd have to do it manually. Other plugins like Social Network Auto Poster (SNAP) or using micropub with clients like OwnYourGram or OwnYourCheckin have intercompatibility with Syndication Links and those URLs will be sucked in automatically.

If you find issues like this, please file them in the github issues repositories for the appropriate plugin so that the issues are known and can potentially be fixed. (Naturally if you can code well enough, you may be able to patch them with a pull request too.)