How does the Helm of Brilliance interact with Great Weapon Fighting?

The item Helm of Brilliance includes a feature that reads:

You can use an action and speak a Command Word to cause one weapon you are holding to burst into flames. […] When you hit with an Attack using the blazing weapon, the target takes an extra 1d6 fire damage. (DMG, 173)

Great Weapon Fighting reads:

When you roll a 1 or 2 on a damage die for an attack you make with a melee weapon that you are wielding with two hands, you can reroll the die and must use the new roll, even if the new roll is a 1 or a 2. (PHB, 72)

This question deals with what dice are affected by Great Weapon Fighting and it seems that the the feature was only intended to affect the weapons damage dice not extra dice from features like Sneak Attack or Divine Smite, or spells such as Hex. It seems that all of the weapon’s damage die, however, are affected and so weapons such as Frost Brand have their additional dice modified (1d6 cold damage in the case of Frost Brand).

In the case of Helm of Brilliance‘s flame effect does is the additional 1d6 fire damage affected by Great Weapon Fighting?

On one hand it seems like no because it is clearly a spell like effect rather than a natural part of the weapon. On the other hand the effect clearly modifies the weapon rather than the attack and so the additional damage could be considered a quality of the weapon.