Princess Astrid has no qualification on her passport. Princess of Belgium is in the surname field in both French and Dutch.
Prince Lorenz is he same. He no longer has Habsburg in his surname on his passport, just his Belgian title.
I don't know about Luxembourg's passports.
I seriously doubt any Habsburg has HI&RH on their passports because their titles are no longer legally recognized anywhere. The would all have Habsburg-Lothringen with some spelling variations depending on the county issuing the passport. (Lorena, Absburgo, etc.).
Some countries do put the HRH or whatever on the passports. I know Luis Alfonso de Borbón has both Spanish and French passports. In Spain, he is simply:
prenom:D. Luis Alfonso Gonzalo Victor Manuel Marco
surnom:de Borbón y Martinez-Bordiu.
But in France he is
prenom:Louis Alphonse Gonzalves Victor Emmanuel Marc
surnom:de Bourbon, SAR le duc d'Anjou
I don't know if France extends this courtesy to
royals other than the Bourbons and the Orleans.
I should mention my knowledge of passports come from 30 years (this year!) working in or with the airline industry. Of course I know specifics about certain royals because it was mentioned in magazine articles or in reference books explaining a title. For example, Luis Alfonso's passports were shown in Point de Vue several years ago.
Do Princesses Astrid of Belgium and Marie Astrid of Luxembourg actually have this qualifier, as well as the Archduchess title, on their passports?
The fact is that all the empires of Europe got dissolved a long time ago; so legally speaking, there are no more Imperial Highnesses around. Even the titles themselves are only unofficial, respected largely in social circles. To the best of my knowledge, only titles in reigning monarchies are legal, and can be officially used on passports.
That being said, Liechtenstein is willing to respect the royal rank of the wife of Hereditary Prince Aloys, since that's what she was born to (notwithstanding the fact that the house of Wittelsbach got deposed a century ago). As such, she is styled as HRH Hereditary Princess Sophie.
But the Benelux princesses are adopting the higher titles and qualifiers of husbands who belong to non-reigning foreign dynasties. That's why I question how their native countries deal with them.