[RPG] Is the Polearm Master Feat compatible with the Two-Weapon Fighting style

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Can Polearm Master make a quarterstaff act as a double weapon?

Back in 3.5e, using both ends of a quarterstaff to attack was treated as using a "double" weapon, which counted for all intents and purposes as if the character were dual-wielding. 5e does not have "double" as a weapon property, but it does have the Polearm Master feat, which allows a character to make one attack at d8+Str and a second attack at d4 using your bonus action. Mechanically, this is identical to a character using a Longsword in one hand and a dagger in the other (which would require the Dual Wielder feat).
My question is if a character wishes to wield a single quarterstaff, could they benefit from the Two-Weapon fighting style that is presented in some character classes and add their Str to the d4 damage when they attack with the opposite end of their quarterstaff?

The RAW seems to say that you specifically can only dual-wield by having a separate weapon in each hand, but is there any word on RAI for this?

Below are the relevant excerpts from the PHB:

Normal Two-Weapon Combat

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

(PHB 195)

Two-Weapon Fighting Style

When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack.

(PHB 72)

Polearm Master Feat

You can keep your enemies at bay with reach weapons.

You gain the following benefits:

  • When you take the Attack action and attack with only a glaive, halberd, or quarterstaff, you can use a bonus action to make a melee attack with the opposite end of the weapon. The weapon’s damage die for this attack is a d4, and the attack deals bludgeoning damage.

  • While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.

(PHB 168)

Dual-Wielder Feat

You master fighting with two weapons, gaining the following benefits:

  • You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.

  • You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.

  • You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

(PHB 165)

Best Answer

I'm not sure if I got your question right but

You already apply the ability modifier to the 1d4 attack with Polearm Master.

This is from the sage advice compendium:

Polearm Master Can I add my Strength modifier to the damage of the bonus attack that Polearm Master gives me?
Yep! If you have the feat and use the Attack action to attack with a glaive, halberd, or quarterstaff, you can also strike with the weapon’s opposite end as a bonus action. For that bonus attack, you add your ability modifier to the attack roll, as you do whenever you attack with that weapon, and if you hit, you add the same ability modifier to the damage roll, which is normal for weapon damage rolls (PH, 196). A specific rule, such as the rule for two-weapon fighting (PH, 195), might break the general rule by telling you not to add your ability modifier to the damage. Polearm Master doesn’t do that.

You don't need the Two-Weapon fighting style to do that, it's already incorporated in the feat.

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