The damage from Unarmed Fighting is not an attack, and only attacks are made magical.
The Insignia of Claws says:
Such attacks are considered to be magical.
“Such attacks” here refers to your unarmed strikes and attacks made with natural weapons. It only makes your attacks magical. Features that are not attacks are not made magical. Since the damage from Unarmed Fighting is not an attack, it is not made magical by the Insignia.
That said, there’s nothing wrong with a DM allowing the damage to be magical. Nothing about the game is going to break.
Monster resistances are inconsistently worded.
For some monsters with resistance, it won’t matter if the Unarmed Fighting damage is magical or not, since it is not an attack, and many monsters have resistance to bludgeoning only from nonmagical attacks. For example, an Air Elemental;
Damage Resistances Lightning, Thunder; Bludgeoning, Piercing, and Slashing from Nonmagical Attacks
However, some monster resistances do not discriminate the source of the damage. For example, the Archmage’s resistances are:
Damage Resistances Damage from Spells; Nonmagical Bludgeoning, Piercing, and Slashing (from Stoneskin)
So if a monster’s resistance specifies attacks, then the Unarmed Fighting damage bypasses it whether or not you rule it is magical. If the resistance does not specify attacks, then it only bypasses resistance to nonmagical bludgeoning if you rule that the damage is magical.