"Aku" should be okay too, no?
Yes. They both has the same meaning, with "Aku" for casual conversations and "Saya" for formal conversations.
Is "ngomong" not acceptable for berbicara?
It should be. But maybe it's not accepted because it's informal
Is saya bicara okay?
It is actually okay, too.
Duolingo (and the moderators) should know and accept this.