I'm French and I follow information sources that strive to be neutral or lean towards New Caledonian autonomy or independence, and this is the first time I hear this claim.
Unfortunately, it's hard to prove a negative, so my answer will be mostly showing sources that you'd expect to mention this, but don't.
Vocabulary note: Kanaky is the endonym for New Caledonia. Kanak is the name of New Caledonian aboriginal peoples.
I will often cite Le Monde, generally considered to be the French newspaper of record. They tend to lean center-right (i.e. the same orientation as the French government at the time), but they do not refrain from publishing things that would be critical to the government. Links to Le Monde website may be accessible to subscribers only.
The French government deployed military means
True, but very limited. The army was deployed to secure the main airport and sea ports. This was reported in Le Monde on 2024-05-16. There are a few more details in a 2024-05-17 article on France 24 (France 24 a public channel, so they might be hiding facts that are unfavorable to the government, but you'd expect competing media to call them out). A search on Le Monde for “Nouvelle-Calédonie armée” dated 2024 does not turn up other relevant articles.
In addition, the gendarmerie is deployed. Despite having military status, this is a police force: basically the gendarmerie is the police in rural areas and the police is the police in urban areas.
and excessive use of force
This is subjective so I won't investigate this. Generally left-leaning media are making this claim while right-leaning media consider the government's use of force justified.
which led to several deaths among the Kanaks
This has been reported. For example an article in Le Monde dated 2024-08-15 reports the 11th death since the beginning of the “troubles” on May 13.
169 injured, 2235 arrests
There have been many reports of injured people and arrests, including 1 2 3. I can't confirm the figures (and the exact time period is not given anyway), but they're plausible. An official report from the representative of the national government claims 2343 arrests as of 2024-08-23. I don't particularly expect the government to hide arrests because they want to counter the extreme right's accusations of being too soft.
Incidentally, these numbers confirm that the claim is about the 2024 events, and not some long historical period.
including hundreds of arbitrary arrests and detentions
Loyalists claim that the arrests are justified. Independentists claim that the (or at least many) arrests are political. I'll leave that one to the historians.
and more than 500 victims of enforced disappearance.
I can't find anything in Le Monde about disappearances. A search yields nothing relevant (I tried a few other variations as well; if you search in French, note that “disparu” literally means “vanished” but is also a common euphemism for “dead”). I do not recall any disappearances mentioned in the news media that I follow.
The CCAT, a pro-independence organization that lead most of the recent pro-independence movement, held a congress on 27–28 July and published a declaration whose complete text I unfortunately cannot find. Secondary sources such as Le Monde, La Première (government channel), Radio 1 in French Polynesia (I don't know their political leanings) give partial information. They report that CCAT made a number of demands, including the departure of police reinforcements, investigations into past actions by the police, and the return of prisoners who have been transferred to mainland France. Nothing about any disappearance.
I looked at blogs of some pro-independence movements:
All of these support independence, and criticize the repression from the French government. They have demands, including the return of the prisoners transferred to mainland France. I couldn't find anything about a disappearance.
A note on prisoners transferred to mainland France: this has been widely reported (Le Monde, Mediapart (Mediapart is a French left-wing newspaper), L'Humanité (L'Humanité is a French left-wing newspaper), …) and is controversial. Seven CCAT leaders were arrested in New Caledonia and transferred to mainland France. “Enforced disappearance” might plausibly a deformation of the fate of these people — they have been forcibly taken away from New Caledonia, but everyone knows where they are. But anyway all sources agree that it's 7 people, not 500.
Since I cannot find any news site or a pro-independence source that reports “500 victims of enforced disappearance”, I conclude that this part of the claim is bogus.