Your PC can invent a new technique on his own

Taken from wikipedia's entry on the Southern Praying Mantis kung fu style:

Praying Mantis

The association of the term "Praying Mantis" with the style is also controversial. Each branch of the style offers a different explanation.

The traditions of the Chow Gar and Kwong Sai Jook Lum branches each maintain that their respective founders Chow Ah-Nam and Som Dot created their styles after witnessing a praying mantis fight and defeat a bird. Such inspiration is a recurring motif in the Chinese martial arts and can be found in the legends of Northern Praying Mantis, both White Crane styles, T'ai chi ch'uan, and Wing Chun.

If your rogue is observant enough - he may become inspired by some natural phenomenon, gaining insights about efficient and effective motions useful for offense, defense and maneuvering.

He doesn't need to be an acetic philosopher (though he may grow to become one as he advances) - he just needs the conviction and self-discipline to push him to go through the physical training required to hone his technique.

If your group's play style allows placing more emphasis on narrative at the cost of poorer combat optimization, you can even select to make the transition of your PC's reliance on his new martial skills gradual - continue wearing armor, combine armed attacks with unarmed strikes (not necessarily in the same round, though), etc. But, eventually your PC will start feeling that his armor is limiting him, preventing him from escaping blows that using his technique he can now avoid - then he may consider giving up on wearing armor. Same goes for unarmed strikes - have him hold on to his dagger, but occasionally use a flurry of blows⁽¹⁾ using a fist and an elbow of his other, empty hand - and let him realize that his hands have become faster and more accurate and than any weapon he is used to. This can lead him to relinquish the use of melee weapons, to use an interchanging combination of weapon and unarmed strikes or perhaps even seek "better" weapons which are more compatible with his developing abilities.

Bottom line - just because the "metagame concept" says your PC can now do something, doesn't automatically means he is aware of that and trusts himself enough to give up his old ways completely and immediately.

Or, he can find written records of a technique

Taken from wikipedia's entry on the Northern Praying Mantis kung fu style:

Origins

There are many legends surrounding the creation of Northern Praying Mantis boxing. One legend attributes the creation of Mantis fist to the Song Dynasty when Abbot Fu Ju [...] supposedly invited Wang Lang and seventeen other masters to come and improve the martial arts of Shaolin.
The Abbot recorded all of the techniques in a manual called the Mishou ("Secret Hands")
[...]
This manual supposedly disappeared until the Qianlong reign era,
[...]
The manual records Wang Lang "absorbed and equalized all previous techniques" learned from the 17 other masters.

Aside from self mastering a technique, your rogue may find a documentation of a certain technique - this can come instead of having him invent it from scratch, or as a latter supplement giving him more inspiration and advanced abilities.

At either case, you can probably work the narrative details with your GM so they won't clash terribly with the campaign settings (assuming that having a monk in it doesn't do that in the first place).

(1) Or whatever equivalent term used in 5e for that...

If I may indulge in a frame challenge.

A group of wandering murderers who, without provocation, attack an intelligent sentient being deserve what they get. This is especially so since, earlier in the module:

There is an evil Red Wizard of Thay who the party is expected to deal with in a non-violent fashion.

There are any number of ways that the party could interact with the dragon to their mutual benefit, albeit not without risk. From the Monster Manual p.95 (my emphasis):

The most cunning and treacherous of true dragons, green dragons use misdirection and trickery to get the upper hand against their enemies. Nasty tempered and thoroughly evil, they take special pleasure in subverting and corrupting the good-hearted. In the ancient forests they roam, green dragons demonstrate an aggression that is often less about territory than it is about gaining power and wealth with as little effort as possible.

Green dragons are consummate liars and masters of double talk. They favor intimidation of lesser creatures, but employ more subtle manipulations when dealing with other dragons. A green dragon attacks animals and monsters with no provocation, especially when dealing with potential threats to its territory. When dealing with sentient creatures, a green dragon demonstrates a lust for power that rivals its draconic desire for treasure, and it is always on the lookout for creatures that can help it further its ambitions.

A green dragon stalks its victims as it plans its assault, sometimes shadowing creatures for days. If a target is weak, the dragon enjoys the terror its appearance evokes before it attacks. It never slays all its foes, preferring to use intimidation to establish control over survivors. It then learns what it can about other creatures' activities near its territory, and about any treasure to be found nearby. Green dragons occasionally release prisoners if they can be ransomed. Otherwise, a creature must prove its value to the dragon daily or die.

There are plenty of levers here that a socially adept party can manipulate. It the party can give this dragon what it wants then they are in a position to make a deal.

Even if the situation deteriorates to violence, it is unlikely to result in a TPK since green dragons don't do that! At worst it will kill one or two, capture as many as it can and scare the rest off. The adventuring opportunities in being a prisoner/slave of a green dragon and having to convince it of your worth each and every day are huge!

Oh, and Phandelver is deliberately set up as a sandbox campaign - your players should know, because you tell them, that in this type of campaign they will encounter things that they can't kill and that will kill them. The appropriate response to hearing about or finding signs of a dragon is to go somewhere else!