I am not sure, but was not there a work profile app in fdroid (possibly by the name of shelter, have not used it for a long time though, so not sure). If is/was on fdroid, it is likely to be foss, and not have these things.

Also, unless the app has internet permisions, I would not worry much about it. Apps can send data even without internet permissions (by means of using a 3rd party, like gms (google play service) or something else), but if your app relies on some 3rd party app, i would not consider it safe to begin with.

edit: yes the app is called shelter - https://f-droid.org/en/packages/net.typeblog.shelter/

the app has not been updated for 1.5 years, but readme suggests that it is in a functional state, and just requires maintainance on newer sdks, which they suggest that they still provide. - https://gitea.angry.im/PeterCxy/Shelter. There issues do seem to be active (https://lists.sr.ht/~petercxy/shelter), so I would not worry about that either.