Their eldest child is named Anna-Astrid, for both her grandmothers. But because of the used hyphen this made me think if this was also considered to be 1 one name (similar to for example Willem-Alexander). If that is the case she has, just like her younger brother Maximilian, no middle name (just 1).
The daughters of princess Astrid are called Laura, Luisa and Letizia, despite the "Maria" before/after the hyphen. Yet they have middle names. I guess Maximilian is having the name d'Aviano to in his middle names, like every habsburg male. They look quite traditional and catholic so I think the middle names are not published but still existing.