On the same link you provide in your question, Marvel Wikia - Daken (Akihiro) (Earth-616), there's additional detail under his powers/abilities that describes his retractable claws. One note about his claws is partially pertinent to your question:
Note: Originally possessed three claws on each arm, one claw each from the underside was laced with the alloy of the Muramasa Blade, which was excised from him. leaving him with two claws on each arm, but as of recently shown he might have regrown his missing claws.
He started with 3 claws, but one was modified to apply the Muramasa blade. He no longer appears to have the 3rd claw after an encounter with Wolverine.
That said, while he did have the 3rd (underside) claw, there was consideration given to the question you pose about healing. From The Full Wiki - Daken (Earth-616):
The Tinkerer, who bonded the broken piece to Daken's claws, has also implanted artificial housings that act as sheathes inside Daken's wrists that are made of Adamantium. According to the Tinkerer, the substance the Muramasa blade is composed of would severely damage Daken's soft body tissues, hence the presence of the Adamantium sheathes.
Unfortunately, neither of these particular sources are cited, however I believe this scene, which appears to come from Wolverine Origins #36, is the one described above between the Tinkerer and Daken.