I'll split this into "how" could this happen and "what" would happen.
I am not sure how the French will do much better in this than they did in OTL. By this point the Chinese had figured out they could use their numbers advantage to wear down anyone through attrition. The French had a lot to lose and the Chinese could have escalated things by helping Vietnamese rebel. So you can see that the war in OTL ended well for the French. They got a bunch of concessions and things didn't escalate into a nightmare.
Not sure I completely agree with you. China got destroyed militarily, with its ports burned and very little to show on land as well. And the Vietnamese did rebel, quite heavily but got crushed repeatedly
However, the couple victories they had (Lang Son retreat) had a big effect on the French government, making Ferry fall. So to me, it's not as much that France couldn't have gotten Taiwan/Hainan if it really wanted it but rather that there was no political will to get it.
As far as I remember, France could have gotten Hainan but got Kwang Tchéou instead, as it gave it the advantage of mainland access to the Chinese market. What would have Formosa or Hainan had? France already had good ports in the region and not enough people to settle.
Now that means that the PoD you need is a stronger French government or, the easiest, no Lang Son telegram.
Now for the "what would happen". If the French have Taiwan than they probably got Hainan too. It was one of their goals. France will have completely upset the balance in the Far East. The USA will be pissed because of the open door policy. Britain will be pissed. You can forget about them joining the Entente. The Japanese are going to be pissed as well. We might see a Franco-Japanese war instead of a Russo-Japanese war. There might even be a Sino-Japanese alliance to contain France in the East. Speaking of Russia, they will probably benefit from this, and can more further into Manchuria while everyone is distracted. The Franco-Russian Alliance probably still happens. Russia will be able to use France's harbors on the pacific.
Agreed there