No he doesn't. For starters this entire "alliance" was from the get-go kinda a farce. Especially seen from Luffy's perspective this was more along the lines of "oooh pirate alliance, seems very pirate-ish and fun, lets do it". Logically the other Strawhats were more serious about it, but in the end if Law would have simply asked him as a friend to help him, Luffy would probably have done the same thing.
All that stuff about a 'position of authority' is simply redundant. Luffy and the Strawhats didn't do all this stuff for the sake of honoring the alliance and the alliance is also only sporadically mentioned just for the sake of keeping up appearances. Law seriously is currently not really giving much about not showing enough authority, he just wants to see DD go down above all else. You are simply giving way too much value to this whole "showing authority" thing, hence invalidating your statement that Law needs to power up.
And that Law would be showing less authority than Sanji and Zoro is simply not true. Law was able to easily give Smoker a run for his money, beat Vergo and just beat Trebol while being half dead. Vergo and Trebol had the same status as Pica who Zoro beat. Sanji on the other hand hasn't made an appearance in ages and got subdued by DD. So where you get the idea that both of them have shown more authority than Law, who has continuously been trying to take on DD, is a mystery to me.
Lastly Law is not a Strawhat, he is not a main character. It's very well possible following recurrent OP patterns that after this arc his original role is over. Oda regularly gives old characters new roles or recurrent appearances, but often they only had one real role. For example Vivi was a main protagonist in the entire Alabasta arc, but after that she just had some recurrent appearances. No doubt Law will be more seen than Vivi after this arc, but that doesn't take away that this arc was pretty much "his" arc. His entire past was shown, what made him go one was thoroughly elaborated on. So there is no reason whatsoever for him to get a power up at this stage, something he doesn't need either way and even less because of a reason like showing authority in a farce-like alliance especially when Law at one point tried to end the alliance.