So many possible excuses he could've made that would've worked to keep his precious accessory.
I guess he just didn't want to lie or something.
Stuff like "It's a cursed item, before I met you my main goal was to break the curse." or "I need this item to suppress a curse on my body, I cannot be separated from it." shouldn't have caused any loss of honor for the girl since it shifts the reason away to a neutral third party.
Or even just the truth would work, since he already knows restoratives in this world are too weak to affect him.
He needs it to maintain his health, and by extension serve as her bodyguard. If he loses it, he cannot heal effectively, and he will die in combat sooner or later.
It's weirdly written that he just clammed up and let it go, after it was painted as having such emotional value to him.