TL;DR - Major image explicitly states that the only control you have over the illusion is to make it move from one spot to another or emit sound, which it will do in a natural fashion. Dodging attacks thus doesn't seem to be within the possibilities of this spell. On the other hand, Project Image and Mislead both allow you to make the illusion "behave in whatever way you choose", which should include more complex actions such as dodging attacks.
Rules as written, it seems that what you seek to do is impossible to perform using Major Image. Simply put, the Investigation check is for observers not physically interacting with the illusion, and the NPC would immediately notice something is wrong when his blade goes through the "monk". From the spell's description, the only actions explicitly allowed for the illusion are moving from one place to another within range and emitting sounds, which the illusion will appear to do in a natural fashion. Complex actions such as evasive maneuvers are not explicitly mentioned in the spell's description, and can't be replicated by simply moving the illusion around, at least not realistically.
Personally, I'd rule that such fine control of the illusory monk isn't possible with major image, especially as you control the illusion from an outside perspective. For an example of how difficult this can actually be, you can watch this video of people playing soccer/football using VR glasses to see from above instead of with their eyes. Note that these soccer/football players are also directly controlling their own bodies rather than mentally ordering an illusory clone around, which intuitively feels like it would be more difficult.
Spells that should have the desired effect.
Project Image (PHB p.270) is a 7th level illusion spell accessible to wizards and bards which allows them to project an illusory clone of themselves up to 500 miles away. As with major image, the illusion is intangible, and it actually dissipates if it takes any damage. However, unlike major image, the spell allows you to "ride" the illusion, spending a bonus action to see through the illusion's senses.
You can use your action to move this illusion up to twice your speed, and make it gesture, speak, and behave in whatever way you choose. It mimics your mannerisms perfectly.
This could be interpreted as the illusion mimics your actual movements, or at least performs the commanded action like you would have performed it. So you could mentally command it to "dodge", and it would dodge to the best of your own abilities. Failing that, you could argue that your physical body performs the dodge and that the illusion mimics your movements.
All of this is moot if your DM allows you to actually act through the illusion while you share its senses, however (as a 7th level spell, it should be cool and powerful). Nonetheless, the possibility to have the illusion "behave in whatever way you choose" implies that dodging attacks should be possible, unlike major image.
As a DM, I'd rule that if the attack would've missed your character (counting only your dexterity modifier to AC, as a hit to the armor would technically go through the illusion and thus reveal it as such), then it also missed the illusion, provided you are currently sharing its senses and actively trying to avoid the attacks. Think "Luke Skywalker vs Kylo Ren", except that you can't afford to get hit.
If your DM is generous, Mislead (PHB p.260) would provide another alternative. As a 5th level spell, it is more easily accessible to the players. Like project image, it allows you to make the illusion "behave in whatever way you choose". But unlike Project Image, the spell description does not actually state that the illusion mimics your mannerism, despite the fact that you can still share its senses to control it. If your DM requires that, only project image would qualify. On the plus side, it doesn't dissipate if it takes damage, but most DMs would likely rule that it would still reveal it as an illusion.
Getting the monk to cast these spells.
The problem with those two spells, however, is that although they seem to be just what you need, they can only project an image of the caster. As such, you'd need a way to have the monk cast it himself if you want your original plan to work.
A ring of spell storing (DMG p.193) allows a spellcaster to store up to 5 levels of spells in a ring, which anyone can then attune (no spellcasting requirement) to cast the spells stored within as if they were the original caster. While this wouldn't work with Project Image (7th level spell), it does work with mislead (5th level spell), provided that your DM accepts that Mislead functions the way you want it to function.
A spell scroll (DMG p.200) would also allow the monk to cast the spell, as long as you can find a way to get Project Image on their spell list. A single level dip in the wizard or bard class, though not exactly efficient for most monks, would work just fine, provided that they can succeed an Intelligence or Charisma check (DC17 or DC15) when casting the spell. To reduce the chances of failure, cast guidance and enhance ability on the monk before he attempts to read the scroll, so he gets a 1d4 bonus on the roll as well as advantage.
Finally, potions exist for other illusion spells such as Invisibility. If your DM allows you to brew your own potions, the party's wizard could create a potion of Mislead or a potion of Project Image, which the monk could then drink at the appropriate moment. The process for creating such a potion would be up to your DM, but it would technically be possible.
In the end, the hardest part of this plan (other than convincing your DM that these spells function the way you want them to) is to find a way to have the monk cast the required spells. If you can manage it, however, I would definitely allow it as a DM, even allowing the monk to add his wisdom modifier to his AC when "riding" the illusion, as he's technically sharing its senses and reacting as if he were there.
Were I your DM, I'd definitely allow project image to work as you plan it to, but probably not mislead, as it does not specify that the spell mimics your mannerism perfectly.