dnd 5e – Can I use the martial arts function to change the 1d4 of Kensei Shot to my martial arts death?

The kensei weapons are monk weapons.

Kensei weapons. Choose two types of weapons to be your kensei weapons: a melee weapon and a remote weapon. (…) The weapons of the chosen types are monk weapons for you.

All the rules that apply to Monk weapons also apply to the weapons chosen by the Kensei.

However, the damage of Kensei's shot is additional damage in addition to the normal damage of the weapon. You can not improve it to use your martial arts die., since the martial arts die only replaces the normal damage of the weapon.