Kensei Shot when the weapon has multiple damage types?

Question

I was allowed the Storm Boomerang (Princes of the Apocalypse) on my Kensei Monk character and decided to make it my Kensei ranged weapon. But I am unsure if the Kensei Shot ability would add 1d4 for each damage type the weapon deals or just one, and if only one, which?

Kensei's Shot. You can use a bonus action on your turn to make your ranged attacks with a kensei weapon more deadly. When you do so, any target you hit with a ranged attack using a kensei weapon takes an extra 1d4 damage of the weapon's type. You retain this benefit until the end of the current turn.

Storm Boomerang

Weapon, range, uncommon

Simple weapon, 1d4 bludgeoning - range 60/120 ft.

This boomerang is a ranged weapon carved from griffon bone and etched with the symbol of elemental air. When thrown, it has a range of 60/120 feet, and any creature that is proficient with the javelin is also proficient with this weapon. On a hit, the boomerang deals 1d4 bludgeoning damage and 3d4 thunder damage, and the target must succeed on a DC 10 Constitution saving throw or be stunned until the end of its next turn. On a miss, the boomerang returns to the thrower's hand.

Once the boomerang deals thunder damage to a target, the weapon loses its ability to deal thunder damage and its ability to stun a target. These properties return after the boomerang spends at least 1 hour inside an elemental air node.

Answer 1

I think the wording strongly suggests exactly 1d4 total damage, while leaving dealing with multiple damage types open to interpretation.

So when weapon deals two damage types, either DM or player would need to select the type for 1d4 (or split the roll result into multiple damage types).

Also I think there are two wordings in use for such things:

1. "the weapon deals..."
2. "the target takes an extra..."

Personally I would consider "the weapon deals" as a weapon's type, but "the target takes" not (but as an extra magical/mechanical effect). Since boomerang wording is "the boomerang deals bludgeoning and thunder", in my opinion both types are eligible as weapon's damage type.

Some Crawford takes on topic:

A magic item like the flame tongue and the frost brand deal extra damage of a particular type when they hit. That extra damage doesn't change the underlying damage type of the weapon. https://twitter.com/JeremyECrawford/status/956938559205724160

Supports that "extra damage" wording doesn't add as weapon's damage type, however doesn't clarify if "weapons deals" does.

Hunter's mark uses the same damage type as the attack that triggers it. If the attack has more than one damage type, choose one. https://twitter.com/JeremyECrawford/status/808824943299264512

Supports that you would need to choose one type for the roll, if presented with multiple type choices for an effect.

Answer 2

The weapon's damage type is bludgeoning

The Kensei's shot ability refers specifically to the weapon's damage type. In the case of the boomerang, the damage type is bludgeoning, as seen in the text you cited :

Simple weapon, 1d4 bludgeoning - range 60/120 ft.

The part of the description that comes after this does not describe the weapon's inner stats, but the weapon's abilities, in particular the ability to trigger additional lightning damage. The weapon's lightning damage comes from the ability, and is not part of the weapon's damage type. Because of that, only the bludgeoning damage can be amplified by Kensei's shot.

For the lightning damage to be amplified as well, Kensei's shot should have been written as something like "any target [...] takes an extra 1d4 damage for each type of damage inflicted by the attack".