Bard's kinda doing things for the necessity of the universe. I can imagine Bard not being fussed by the Targonians having Aurelion Sol because he has some level of assurity that Aurelion will get free and thing'll be fine. A hundred thousand years is a blink in the life of the universe, and right now the Targonians are doing what's necessary to protect the universe from the Void while also not (seemingly) being a threat to the universe and life within it.
There's also the issue of Bard just being a super busy dude and so he might just not have the time. Protecting the universe is a 24/7 kinda job after all. Alternatively Bard might actually be worried that the Targonians could use Aurelion Sol or their own power to vanquish him, considering Bard is a "killable" entity.
I can imagine Aurelion Sol being mad at Bard for not helping him, as he's a force necessary to the continued existence of the universe, and Bard's just kinda letting it slide. One thing to of course keep in mind though is that Aurelion Sol is another character who doesn't see the full picture, even as a mighty star forging dragon. Some things are likely even beyond him. I personally like to imagine that Bard has a close to perfect grasp of how the universe works, and how things like Nagakabouros even function. While Aurelion Sol might say "Saving me is critical to the universe's survival Bard!" Bard might retort with "whimsical musical sounds with an underlying deep, somber, and serious notes.", and Aurelion Sol would then just be like "...well you make a good argument but still!"
Also Aurelion Sol might not like Bard. They are wholly different beings who exist in different ways. The star dragon might not like the spiritual vagabond.