This is a direct follow-up to this other question of mine and assumes that, indeed, one may wield a Talisman of Pure Good (or a Talisman of Ultimate Evil) as an improvised weapon and apply both the regular melee damage from an improvised weapon plus the Talisman's radiant/necrotic damage if the creature is not of the right alignment.

The Player's Handbook states the following about Knocking a Creature Out:

Sometimes an attacker wants to incapacitate a foe, rather than deal a killing blow. When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable.

Can one decide to strike a maligned foe non-lethally (and therefore knock them out instead of killing them) with a Talisman of Pure Good (or Ultimate Evil)?

My hesitation relies on the fact that the radiant/necrotic damage from the Talisman does not directly come from the melee attack, but rather from the item. I think it might still work, but I am unsure.

Any melee attack can be a knock-out blow.

You've quoted the complete rules on the matter. The type of damage is not relevant, only the source of the attack - melee or ranged. Both a melee weapon attack and a melee spell attack are melee attacks.

Another user asked What is "Force" Damage? In that case, they were asking how to describe wounds, but I made a key point in my answer:

Until a source of damage interacts with something with resistance, immunity, vulnerability, or some other ability that cites an interaction with a type of damage, the damage type is functionally meaningless.

In D&D5E, you can just as easily fire-damage somebody into unconsciousness (via Flame Blade, a melee spell attack) or shock them into submission (via Shocking Grasp), as you can beat them within an inch of their life. You cannot Lightning Bolt or Fire Bolt somebody without the risk of killing them, because neither of those two spells involves a melee attack (one is a save, the other is a ranged spell attack).

The Talisman may not be an attack at all.

While it isn't written that way, I suspect the intention of that passage is when an unworthy character touches the talisman willingly. If you're allowing that feature to be used offensively, by pressing it against the unworthy, you're dipping into homebrew territory anyway.

If you're allowing a character to make a melee attack with a Talisman, then yes, it would work as you describe. A melee attack is a melee attack, and by the rules can be used as a knock-out blow.

No

Your quote specifically states a "melee attack," but the damage from the Talisman occurs after (when they touch it), so it isn't an attack.