Despite being (effectively) the oldest and most prestigious dynasty in Europe, the Habsburgs had only one marriage alliance with the Romanovs of Russia. Now I can fully understand the reasons why; first both dynasties were very devote and any bride would be expected to convert to her new country's religion and second, the only Habsburg-Romanov match that happened (Tsar Paul's Grand Duchess Alexandra Pavlovna to Emperor Franz's brother Archduke Joseph, Palatine of Hungary) was a miserable marriage that ended with the unhappy Alexandra dying in childbirth. But this is what makes it a challenge! Bonus points if the marriage is between the Tsar and an Archduchess or the Emperor and a Grand Duchess.
Not sure about pre-dissolution Holy Roman Empire, but if I'm not mistaken, after the
Austrian empire was proclaimed the proviso was inserted that the emperor's wife
had to be Catholic (i.e. if she wasn't, then she'd have to convert). The Russians were against their grand duchesses having to convert - and AFAIK, besides the girl who married the emperor/tsesarevich, girls marrying into Russia were generally allowed to maintain their religion - Mieche-Miechen being an example (despite her marriage to an Orthdox grand duke, she remained a Lutheran).
Now that doesn't mean Romanov-Habsburg marriages
weren't on the cards:
Olga Nikolaïevna famously refused to marry Archduke Stephan (son of the aforementioned Palatine of Hungary) because she refused to marry a "king with a paper crown bestowed by Metternich". So maybe if the marriage had gone through, it might've made intermarriage more common.
It doesn't meet with your requirements for a Romanov, but the same Elena Ivanovna who married Alexander of Poland was offered in marriage to seal a treaty between the Empire and Russia, either for Philipp the Handsome or his widowered father, Maximilian.
But in general if Russia wanted an alliance with Austria, they didn't do it directly:
Alexei Petrovich marrying the sister of the Holy Roman Emperor's wife.
Emperor Pavel marrying the sister of the wife to the Holy Roman Empire's heir-presumptive
Empress Yelizaveta considering Maria Anna of Saxony (OTL electress of Bavaria) for Pyotr III. Maria Anna was an emperor's niece and grand-daughter.