DP said that what was 'generally known' was fairly complex, meaning there might be some people who only know dry old history of House Targaryen's rise (and subsequent fall if they were paying attention to recent events) and then rumors of some scion leftover causing a stir in the East.
Some might be paying close attention to recent events and know generally enough about who used to rule Westeros, as they are maritime families or merchant houses or some such, maybe some who have even seen our realm. But beyond general local hearsay, know little about Viserys.
And some might actually be mages and adventurers and spies and just generally people with access to local experts, having not only the best of the histories to peruse from, but also their thumb on the pulse of rumors and events occurring as they keep a close eye on things, in addition to the magical lore to make a guess of what "flight of fancy" might have several thousand tonnes of grains of truth to them, and which are patently false or don't adhere much to Viserys' character and clearly displayed MO.
With that in mind, assuming Bu Gai has the resources to keep abreast of things going on westward, and had his own available experts devote considerable resources even for our own institutions into researching what is actually accurate and what isn't, he might actually be fairly well briefed on what we're all about.
Thing is, what we're all about is kind of confusing and daunting. We've established diplomatic ties with two Genie Empires, fight asymmetric wars with fiendish cabals, and have conquered or annexed large regions in a matter of weeks at most.
So the closest metaphor that would make sense for most of the court if not Bu Gai would be Valyria Reborn, though paradoxically, the whole "free all the slaves" and "spread all the magic instead of concentrate it" might be making their heads spin a bit. The fact that it hasn't ended in chaos and death everywhere another source of confusion, since we just freed tens of millions of slaves over the course of a year and a bit. And they're A) staying freed and B) haven't caused the end of civilization. Oh and, trade is booming like never before. And out of every ten rumors about us, five of them are at least about how fuck-off rich we are.
"Rich as a Lannister" was a common saying even centuries prior when Valyria stood at its height.
We are pretty well at the point where tales of Lannister wealth are much less alluded to than the Dragon's Hoard.
