Deity "favored weapon" on special unarmed "weapons" like 1d8 dogtooth tengu turbo beak

I wanted to make a dogtooth tengu champion who's built like an industrial fridge, has a shield in one hand, and since he has the 1d8 dogtooth tengu beak, instead of a weapon he can leave the other hand free to dish out athletics-based suffering like disarm, trip, tiddytwister, whatever you want

So I tried to select a deity who's favored weapon is unarmed, jaws or something like that.

However, it seems I have to have either a basic unarmed attack of d4, OR a simple weapon in order to increase my damage dice.

Do I understand it correctly, that even if I would select such a deity, it would not one-up my attack damage to d10? Because that would be tragic beyond human comprehension, I really wanted to go full woodpecker and impact-drill through everyone with my d10 unarmed while folding up the survivors with one hand like laundry.

https://preview.redd.it/ojlmjvaidrtd1.png?width=679&format=png&auto=webp&s=00bfa597df8d6ce914baedd386f1b4cf73175bae