How could it be affirmed that the majority of the public know the names from the Funimation broadcast? Is there any source stating that? Even if that were true, Dragon Ball is known in almost all the English-speaking world, not only in the US. And that's not including the rest of the world. More importantly, the manga is the primary work, and normally the main source for most information regarding a work of fiction it's the printed work, like most American comic-book characters like Superman, Batman, Wolverine, etc.
These are the key phrases to decide what names to use:
WP:ENGLISH
"Use the most commonly used English version of the name of the subject as the title of the article, as you would find it in verifiable reliable sources (for example other encyclopedias and reference works)." I honestly don't think that reliable sources will be found about this topic, at least not like encyclopedias and reference works, which leaves us with the following policy.
WP:MOS-AM
"Characters should be identified by the names used in the official English releases of the series. If there are multiple English releases, such as both a manga and anime, use the one that is best known and that has contributed most to the work's becoming known in the English-speaking world (usually the primary work)." This is the key one. We have to decide which one has contributed more to Dragon Ball becoming known in the English-speaking world. In my opinion, it was the manga, since Dragon Ball has been known for a very long time thanks to it, and, in the US, the first broadcasts of the anime where unsuccessful while the manga was already a success, and I have the impression that it was this success with the manga that lead Funimation to keep on trying. But again, this is not definitive. We need the opinions of people from other English-speaking countries in order to have a more accurate perception.
If we want accuracy, which is also desirable by other Wikipedia policies, we should use the romanizations given by the author (Toriyama) in his Japanese works and then include the alternatives as stated in WP:ENGLISH. After all, most guidelines aren't meant to be a strict rule. But for the moment, first we have to define which one, manga or anime, helped more in making Dragon Ball known in the English-speaking world. And remember, US TV is not the whole English-speaking world. And if we go by the primary work, it's manga.