Yes, you can apply your Martial Arts die with your Kensei weapons. Actually, you would be able to do so with any monk weapon.
The Path of the Kensei feature makes eligible weapons into Monk Weapons.
P. 34 XGtE
Choose two types of weapons to be your kensei weapons: one melee weapon and one
ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don’t already have it. Weapons of the chosen types are monk weapons for you. (...)
P. 78 PHB
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.
Dueling adds +2 damage to attacks with melee weapons when wielded 1-handed without a weapon in the off-hand.
P. 72 PHB
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Now, is this optimal, or even good in terms of damage? Well, no. That bonus damage only applies to 2 of your 3/4 attacks per round. Action Surge only doubles 2 of your 3/4 attacks. A fighter is throwing out 4 APR minimum at L20, with all the other goodies Fighter gets.
If you use a shield in your off-hand, you can't use your monk features, and a Longsword is no better than a dagger as a monk weapon (both will use a D10). Essentially, you are a monk with +2 damage (on 2 of your attacks), a few extra HP, Action Surge, Combat Superiority Dice, no Empty Body, and 1 less ASI. You miss out on Perfect Self, but it's kind of rubbish anyways.