You can hold a longsword and a dagger and choose which one you want to use*
*However, you can not attack with both your longsword and dagger in the same turn as an action/bonus action. But you can use them each as part of an Attack action with the Extra Attack feature.
The rules around Two-Weapon Fighting require both weapons to have the light property:
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.
So in this build where your monk/rogue is wielding a longsword in one hand and a dagger in the other, they have the options of:
- Attack action with longsword/unarmed strike/dagger and bonus action unarmed strike if available.
Since one of their weapons is not light (the longsword), they won't be able to use a bonus action to attack with the dagger. But they absolutely can use it as part of their Attack action if they have Extra Attack.
You also miss out on that sweet d10
Because you are wielding a weapon in both hands, you aren't wielding that longsword for the versatile d10 damage. That's kind of a primary point for the Kensei (at least until the Martial Arts damage die gets equal to it).
Alternatively, the free object interaction works here as well. You can wield the longsword versatile for the d10 die, but if you have an opportunity to get sneak attack, you can use your free object interaction to let go of one hand off the sword and draw and attack with your dagger. However, if you want to be wielding just the longsword again in two-hands prior to your next turn, you would need to drop the dagger.
Sneak Attack with the dagger is permitted
As you've noted, the dagger does provide everything you need for sneak attack. When the opportunity arises for that damage, then you can choose to attack with it.
There is no off-hand
Since there is no penalty for 'off-hand', other than not adding your modifier to the bonus action attack for TWF (see above quote), this is a completely viable build option.
In 5e, the idea of off-hand is a bit weird. They have transferred the idea of the 'off-hand' to the bonus action attack. Functionally, you are equally proficient with either hand when using your Attack action. It's when you use the bonus action for attacking with a second weapon that you lose your proficiency.
You will still need to make an unarmed strike while holding your kensei weapon to get the benefit of Agile Parry
Just a note (in case you were thinking of taking the Dual Wielder Feat to allow the use of non-light weapons) that you still need to make an unarmed strike as part of your Attack action (and not your martial arts bonus action) in order to activate the AC boost.
At least by the rules as they are written the dueling bonus applies to both attacks. On page 24 of the SRD the Dueling Fighting Style text is:
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.
Note that the bonus to damage is activated immediately. So, if you lose one of your psychic blades and immediately gain another you get that bonus applied.
Best Answer
The damage die of a psychic blade could increase at DM's discretion.
As you mentioned, the psychic blade fully qualifies as a monk special weapon and could benefit from martial arts and other monk features that require a monk weapon.
The damage die of the second attack could be modified by Martial Arts at DM's discretion following the rule Specific Beats General:
Both Martial Arts and Psychic Blades present some specific rules.
I wouldd (personally) rule that Martial Arts could also increase the damage die of this second attack, since the second psychic blade still is a simple melee weapon that is not heavy nor two-handed, and you also have to invest at least 5 levels in monk to increase its damage die, so it wouldn't break anything to allow it.
Other DMs might see the damage die reduction as a more specific rule, disallowing the second attack's damage die from being modified this way. This is also a fine way to rule it.
However, reading the feature RAW, you cannot create psychic blades more than once per Attack action, and you can create one as a bonus action only in specific circumstances.
The feature lets you create one psychic blade when you take the Attack action, not when you attack as part of the Attack action, so features like Extra Attack cannot use more than one psychic blade, and it has been established with this question that this bonus action is not the same as using Two-Weapon Fighting (and Martial Arts by the same logic). The special bonus action of this feature allows you to do something similar to two-weapon fighting or using the bonus action attack of martial arts.