Segmentation mask in an LabelSetImage is named Object-1, Object-2 by default. There is no straightforward opportunity to set developer-desired label names. This can be definitely accomplished by:

for (auto const &[id, labelName] : LabelMap)
{
    mitk::Label *labelptr = LabelSetImageObject->GetLabel(key, 0);
    labelptr->SetName(labelName);
}

Maybe, its better to get this code into the LabelSetImage class as a function eg.
void SetNewLabelNames(std::map&, int activeLayerId)