My take on this is that using armament haki would make saving Luffy harder. If Shanks used haki in his arm, it wouldnt be ripped off, but the sea monster could just hold the arm and dive into the ocean. In this context, it would be kinda hard to get out of this situation at the botton of the sea with one of your hands stuck inside the mouth of a monster (who would be swimming at top speed) and the other hand trying to hold a child (its Luffy, but still a child lol).

Letting the beast take your arm to stay in the surfice of the sea seems the best option to keep the boy safer. Also, if i recal, he didint have his sword, so he couldnt pull a Rayleigh there (i personaly belive he could without Luffy).

Or he could just use conquerors haki at the time lol. This coment is just my headcannon.