The WG assesses what is a threat to the WG. That's why Robin had a massive bounty as a child.
Robin was a Ohara surivivor: the Marines knew full well about her.
Anyway, we know it's the Marines.
Branniew is the head of the office:
Even recently Aokiji's bounty was described as "discussed by the marine HQ":
There are exceptions, but overall the marines are the one who give bounties.
Of course they knew. They've been in control of the world for nearly a 800 years.
You actually think they don't know of the Kouzuki clan, who created the Ponegylphs that lead to the One Piece? Bruh.
I think the gov. knew of the Kozuki, and probably the marine too at the time.
I don't think they knew Oden was Sakuyaki's son or even know what power did he hold or even if he was still a member of the family since he was a fugitive.
It's dumb, but even if X Drake was there, Oda seemed to want Wano to remain obscure for everyone. People didn't know Orochi had taken the throne, didn't know Oden was dead, etc.
Imho, at best they knew Oden was someone from the Wano royal family.