Now I understand at this point the Holy Roman Empire was a shadow of it's former self, barely holding it's practically sovereign internal constituents together, but the possibility still interests me
I'm going to ask this as a question: Are there not still legacy treaties in place between USA and Holy Roman Empire? Stemming from Revolutionary War mercenaries that performed well?
I'm sorry to say the answer here is simply 'no'. The US never established an ambassador to the Holy Roman Empire and didn't establish one to Austria until after the Empire's dissolution in 1838. They didn't even have an Austrian Consul until 1797. Even stretching your definition to 'any interaction with any part of the Holy Roman Empire', the only treaties were the US, Prussia, and some other minor states signing treaties of Amity and Commerce. These were agreements for recognition, trade, and peace and not major diplomatic maneuvers.
The first 'major' interaction between the US and Habsburgs was during the upheavals of 1848-51. The United States supported the reformers, revolutionaries, and Hungarians in Austria. But the Holy Roman Empire was long gone by then.