Oda set up Sanji's role pretty well this arc. From the first attempt at trying to save Momo
to then saving Momo from the execution
and then worrying about Momo after his encounter with King
Sanji knows how important Momo is for the Samurai. Momo is their entire hope of the new future.
King also knows how important Momo is, and knows if they kill him....they win the war.
Momo is one of the cards against Kaido. He is an important asset, and if the Beast Pirates get a hold of him....they win. This is a callback to Dressrosa when Nami told Sanji and Luffy, about the 3 cards they have against Doffy.
If they lose Momo, they lose their advantage against the Beast Pirates. Thus Sanji knows how important this is since he learnt it from DR. He will protect Momo with everything he got and make sure he doesnt fall into enemy hands.
With Sanji protecting Momo, I can see at the end of the arc Sanji getting a small reward with Hiyori
Anyways how I always saw it going was Sanji going to Kinemon, and Kinemon begging Sanji to protect Momo. This still may happen. Because I really want Kinemon and Sanji to share a moment this arc....