I agree with the above comments about the difference in their relative ranks but I think the main obstacle would be that Eugene never showed any interest in marrying anyone ever. Whether that was because he was homosexual or not I couldn't say but he was offered a pretty good bride OTL, one of the Lauenburg sisters who were heiresses to substantial holdings in Bohemia (with the other sister going to another Imperial General the Emperor wished to reward, Louis of Baden who was coincidentally Eugene's cousin), and he refused the match. So that's the first problem. Then as mentioned there's his low rank, the younger son of a cadet branch of the House of Savoy. Also his older brother, the Count of Soissons and head of the family, would have still been serving Louis XIV at the time Maria Antonia was married and what's more Eugene had only been in Imperial service a few years and had yet to really distinguish himself.
As a side note I'm confused as to what was meant by "exploits in the field first on behalf of Austria and then of France" because Eugene never fought for or on behalf of France and was only ever in Austrian service, either against the Ottomans or against the French.