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Do a Way of the Kensei monk's kensei weapons count as monk weapons?

If they do, since the weapon's regular damage can be replaced by a monk weapon die, can the extra damage from Kensei Shot also be replaced by that die?

Kensei weapons are Monk weapons

The Way of the Kensei monk's Path of the Kensei feature (XGtE, p. 34) says, in part:

Kensei Weapons. Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. (...) Weapons of the chosen types are monk weapons for you.

All rules that apply to monk weapons apply to the Kensei's chosen weapons too.

However, the damage from Kensei's Shot is extra damage on top of the normal damage of the weapon:

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 cannot upgrade it to use your Martial Arts die, since the Martial Arts die only replaces the normal damage of the weapon:

You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.