Regarding to the wisdom of having two separate navigation bars… it depends. For me, one domain, one site. And that means that navigation needs to be consistant.
Now, both sites are significatly distinc and having two different navigation bars is truly an advantage for your visitors … then, suit yourself
I think not doing would be confusing for your visitors, consistent navigation is important for UX. You should start thinking about putting your visitors first, not the search bots. Search bots don’t spend money with you.
I’m sure it has already been explained to you previously that individual Pages are ranked, not sites or domains as a whole.
If it’s on the same domain, it is effectively one site, even if you structure different sections of it differently,
And as others have said above, and we’ve explained many times in previous threads, Google ranks individual pages, not sites, and the ranking will depend on the exact parameters of the search query.
I really feel you could save yourself a lot of unnecessary obsessing if you took the time to watch the Google videos on how search works which I’ve pointed you to before.