Re-using the damage type of the weapon is indeed an interesting idea.
This way the player would simply extend the damage of the weapon of its choice, which means player's choice !
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Mixing One-Handed Fighter and Two-Weapon Fighting could work through itemization, but the cost would be a third feat on top of an item (4 prerequisites), which is a serious penalty against Great Weapon Fighting + Great Wepon Master (2 prerequisites)
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Mixing One Armed Master and Two-Weapon Fighting could lead the player to wield a Greatsword in each hand, which seems unrealistic, excepted for a beefy barbarian.
Monkey Grip has the same problem.
https://dnd.arkalseif.info/feats/complete-warrior--61/monkey-grip--1978/ I think such builds are possible, but only with proper fantasy conditions, like: "At 20 Strengh, the Barbarian wielder ..."
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In order to make the item described in the proposal super generic (had Rogue / Ranger in mind) and with the less dependencies as possible, I simply added flat damage on top of One-Handed weapons.
But maybe I should have targeted
Dual Wielded Weapons rather than
One-Handed Weapons, as in its current form the proposal could add 1.5 damage on top of the One-Handed Fighter for players using a single weapon (power creep).