You might be overthinking it. According to Brian Jacques himself, it seems he didn't go into the depths of exactly what traits the mythological/biblical character Asmodeus represents; he just found it among a list of names of Satan and thought it'd be a fitting one to choose for a snake. I found this information in what seems to be an author commentary on a TV series adaptation:
"How do you come up with the [characters'] strange names?"
"Well, names you can make up. Names you can find. Eh, if I am traveling on a train I'll take a pencil and a scrap pad and I can invent names. Names are easy for the baddies, y'know? Fattybelly, Snottynose. But, some of the other names you will find. Like the name 'Asmodeus'. I was looking in the Bible... it had the names of Satan... all the names of Satan. It had... eh... Lucifer, Beelzebub, Lord of the Flies, Prince of Darkness. And then it had 'Asmodeus'. And I thought, 'Isn't that a wonderful name for a snake, with all those S' in it?' Asssssmodeusssss. So, that was a name I found. But, you invent a lot of them yourself."
Further analysis can be found in the master's thesis From Wanderer to Warrior: Martin’s Journey to Sainthood in Brian Jacques’s Redwall Series (Marie Bleimeister, Georgia Southern University, 2019), a spiritual (Christian?) analysis of Redwall:
It is not surprising then when the time comes for Matthias to stand against the adder Asmodeus that his own strength deserts him and is replaced by fear as he stares into the eyes of Asmodeus. In the Bible, the devil goes by many names, one of which is Asmodeus. Jacques comments that “I was looking in the Bible... it had the names of Satan... all the names of Satan. It had..eh...Lucifer, Beelzebub, Lord of the Flies, Prince of Darkness. And then it had ‘Asmodeus’” (“How Do You Come Up with the Character’s Strange Names?”). In addition, the Catholic Encyclopedia notes that Asmodeus means “‘to destroy’: so that the being would correspond to the demon called Abaddon, the Destroyer in the Apocalypse 9:11” (“Asmodeus”). Marian Copeland in her article “Crossover Animal Fantasy Series: Crossing Cultural and Species as Well as Age Boundaries” notes that “careful readers will find Asmodeus also defined as one of nature’s ‘efficient undertakers’” (292). Matthias has entered the lair of a devil and to return to the light, he must retrieve Martin’s sword. At this moment, the spirit of Martin comes to Matthias and says,“I am that is! Matthias, why do you sleep! There is a warrior’s work to be done here! Pick up your sword, Matthias!... Strike out for me now, my brave young champion!” (Redwall 320). Martin comes to break Matthias from the power of Asmodeus and to bring him up from the darkness. Martin becomes Christlike and rescues Matthias from the pit of Asmodeus. Asmodeus traps Matthias with his persuasive ways, and Martin must pull Matthias out of his trance into the light to be transformed by Martin.
The warrior must be ready to battle those who are considered a threat to the earthly paradise of Redwall Abbey. Blackford notes that “Matthias’s rebirth into a larger-than-life status once he has fought the adder suggests his participation in a cosmic, archetypal battle of good against evil, aligning his triumph with the triumph of God over the adder, who the narrator describes as a pathway to eternity, and the pagan Cluny, ‘God of War’ (Redwall 8)” (98). In other words, Matthias’s victory over Asmodeus is, in fact, a victory over the devil. Therefore, his defeat of the devil aligns him with God’s righteousness. In addition, Blackford remarks that because Jacques describes Matthias as “larger than life” and “suddenly reborn” that Matthias has in effect “become a divine being, reborn from his travel into the underground (hell) of the adder’s den, suggesting not a wise regard of the sword but an appreciation of his own power to penetrate the very earth with the sharp point of his weapon” (98). Indeed, Matthias does relish his own power, forgetting that the power comes from Martin the Warrior. However, Matthias is not divine, only divinely inspired by Martin, though at this moment, Matthias may believe he is a divine being. As noted before, it is not until Matthias is an adult mouse that he becomes wise in regard to the sword.
There's a lot of analysis and discussion here, but the relevant part with regard to your question is that Asmodeus is seen as representing the Devil (whom Matthias overcomes, with the divine help of Martin, in order to achieve his new [saintly?] status), and Asmodeus is one of the names for the Devil. Whether or not you buy into this theory of Christian symbolism, at least the character of Asmodeus being named after the Devil is plausible and makes sense.