Although I consider myself relatively well-versed in Polish history in the 1900s, my knowledge of the Commonwealth period is incredibly sparse. 'Balance of power' politics was beginning to take root during the late 18th century, and it seems logical to me that Britain or France would want to guarantee the borders of Poland-Lithuania in an attempt to stem the aggression of its expansionist neighbors. I don't know very much about the political landscape of the time but I'm still surprised that the Commonwealth was left completely friendless during the Partitions.
Even during the First Partition, could Polish-Lithuanian troops put up any semblance of a fight either way?
it seems logical to me that Britain or France would want to guarantee the borders of Poland-Lithuania in an attempt to stem the aggression of its expansionist neighbors. [...] I don't know very much about the political landscape of the time but I'm still surprised that the Commonwealth was left completely friendless during the Partitions.
I think you quite overestimate Western powers interest in Polish-Lithuanian Commonwealth. We can take War of the Polish Succession (1733–1738) as example- nor France, nor Sardinia, nor Spain had much interest in who will be new King/Grand Duke. They started war just to strike against Austria. And lost, it must be added. If France (and allies) were unable to defeat Austria (and allies) in Rhineland and Italy, much less they could do about intervention in Commonwealth itself- even if they really wanted to.
Even during the First Partition, could Polish-Lithuanian troops put up any semblance of a fight either way?
There was not much of troops to speak of. I don know much about Poland, but Lithuanian army counted less than 5000 mens before Four Year Sejm reforms (1788-1792). Four Year Sejm decided to increase Commonwealth army up to around 100 000 soldiers (66 000 for Poland and 33 000 for Grand Duchy) but despite effort these numbers were never reached- Grand Duchy had about 17 500 mens in 1792. About 1 500 can be added if we count Radziwill militia (a private army, formed primarily for protection of Radziwill properties but it was a considerable force, capable to go face-to-face against Russian regular army). Compared to forces of partitioning empires (Russia send force of 100 000 forces in war of 1792 alone), Commonwealth had no real military force to counter, even if ignoring political chaos that undermined Commonwealth ability to resist (things like Targowica confederation).