Necessary but insufficient
It would help. It would more than help; I have a hard time imagining them being functional without it. The necklace of natural attacks or scorpion kama are generally necessary for Monks. That said, the necklace of natural attacks does exist (as do similar items in Pathfinder), and it’s not nearly enough to make the Monks good.
The Monk’s problems are problems of design: the people who wrote it evidently had no clear idea of what a Monk was or should do. Thus it receives a mish-mash of random abilities that do not synergize (and frequently contradict one another, see Flurry of Blows and Fast Movement).
To “fix” the Monk, one must first come up with a clear vision of what the Monk is supposed to be and do, and then most likely rebuild it from the ground up focused on that vision.
Or simply use one of the many classes that can effectively model one or more possible visions of what a “monk” should do, without any levels in the “Monk” class. The Cleric and Psychic Warrior are both Open Game Content and quite capable of fulfilling most roles you could imagine for the Monk, for instance. I’d argue that it’s entirely reasonable to treat a Barbarian’s Rage as “Zen Focus” and waive the non-lawful requirement (which I’d further argue is dumb to begin with). Such a “Barbarian” multiclassed with Fighter for combat maneuver mastery and perhaps taking Improved Unarmed Strike would make a decent monk. And if you have Tome of Battle, the Swordsage also does an excellent job.
No.
The key here is that the Monk weapon ability doesn't actually add the keyword "finesse" to the weapon, it just lets you use Dex for attack and damage.
Defensive Duelist specifies that a weapon have that keyword, and thus, no, it can't be used.
Adding more fuel to this fire is Jeremy Crawford talking about natural weapons and coming down on the side that just because you are attacking with Dex does not mean you are adding finesse:
No. A weapon has the finesse property only if its description says so, and using Dex. to hit doesn't equal finesse.
Would this be a game breaking houserule? No, that's quite unlikely. The Staff does d6 or d8 damage, and there is already a finesse weapon that does d8 (Rapier). The monk isn't proficient in that, so it's not available, but it's not out of bounds for them to be able to use it.
Note: Brian points out that two monk weapons are finesse weapons, and as such would be usable with this combination (and get the Monk's proper monk weapon die). These are the dagger and the short sword.
Best Answer
Yes, you can
Crusader's Flurry (from Ultimate Combat) is a feat commonly taken for that purpose, it will allow your character to take a weapon, related to your chosen deity, and treat that weapon as a monk weapon for all purposes.
Do note, however, that it has heavy requirements to base your build around. It requires both channel energy and flurry of blows. While this could be obtained by multiclassing, you are probably combining two medium BAB classes, which will hurt you heavily and not really justify the combination just to take the feat.
The Sacred Fist warpriest archetype will grant you both requirements, but you still gotta waste a feat on Weapon Focus, and probably another on the weapon proficiency, as the archetype loses both Weapon Focus and proficiency on all simple and martial weapons, substituting their proficiencies to a list similar to the monk's (which is pretty bad). The weapon proficiency has better ways to obtain, such as using a trait or multiclassing into a full BAB class, which is probably a bad idea as it will slow your spellcasting progression.