Why did the dispute between Spain and France over the Mexican border finally end?

Answer 1

After the 7 Years War, called the French and Indian War in what is now the USA, the French ceded their claims to the Louisiana Territory to Spain [Treaty of Paris, 1763]. Any disagreement at that time about the border would have ceased.

The specific answer to your question is that the dispute ended by the Treaty of Paris that ended the 7 Years War.

In 1800, France regained the Louisiana Territory in the Treaty of San Idelfonso and napoleon sold the French claim to the USA in 1803.

Answer 2

When Mexico became an independent country would be my guess.