There is actually some really strong evidence, it’s posted waaaay back on this blog though. The collar is the crucial thing because both Iroh II and Bumi have a royal Firenation colour. You would expect them to have the same collar seeing as they are wearing similar uniforms, but the fact each of their collar’s has the royal design on it (confirmed by looking at pictures of Azula and Zuko in ATLA) really hints that Bumi married into Fire Nation royalty.
Iroh II obviously didn’t call Bumi “dad”, but then again that situation never presented itself. I think it’s just a case of wait and find out.