Still haven’t had time to test this, but rechecking everything I think I now better understand what is happening.
My guess is that you are losing your “Vermessung” project configuration by using this workflow. When you directly open a file, QField is probably creating an ad-hoc project with some default configurations, especially on some less trivial settings such as attachment path definitions (such as relative/absolute and other details). When you open the “Vermessung” as a “Individual Datasets Base Map”, you most likely aren’t bringing the whole project “plus some layers”; it sounds reasonable that the ad-hoc project’s settings get precedence over the Base Map’s because the function might have been designed to provide just a “custom background”.
The way you describe you are using the “Individual Datasets Base Map” sounds more like a workaround to dynamically add layers to a project on QField, and not just as a way to have some custom background to accompany the isolated dataset. If this is the case, sadly it is a big current limitation of QField; if you want to add layers to a project, you have to do it on QGIS.
If you need this functionality, consider adding your voice on the Ideas platform so that the developers can know there is demand for features that address this issue. For example, there currently is this request to add the ability to create new layers. Maybe an alternative to what is proposed there is to be able to just “copy” a layer into an existing project; perhaps an option to add it as a temporary/scratch layer can be helpful to avoid polluting the actual project. Scratch that, I keep forgetting about this plugin by the man himself! Maybe that can be of help.