Searching on google, I found this :

« La Belle France » est un pamphlet de Georges Darien publié en 1900. C’est probablement l’un des pamphlets politiques les plus puissants et les plus violents que nous connaissions, tout là-haut avec Napoléon le petit de Victor Hugo, avec le Discours de la servitude volontaire de La Boétie. « La Belle France » n’est pas un pamphlet, c’est un exercice de tauromachie. « La Belle France », c’est la mise à mort de la Belle Epoque.

→ « La Belle France » is a lampoon of Georges Darien publicated in 1900. It is probably one of the most powerful and violent politic lampoons we know, with Napoléon le Petit from Victor Hugo and with Discours de la servitude volontaire from La Boétie. « La Belle France » is not a lampoon, it is a bullfighting exercise. « La Belle France » is the killing of la Belle Epoque.

This article, if you can read French, explain about what the text is and from where, I think, is coming the expression you describe.

We can see on ngram that this expression is existing already, but maybe not with this negative connotation. I believe G. Darien has used an already existing expression to criticize it and all what it represended.

A more careful search with ngram finds the phrase being used by La Fontaine … I’m not saying he invented the phrase, maybe it was a commonplace in speech even in his time.

Is it well-known in France too (I see it’s a brand name) or is it mostly an English expression?

The brand name is anecdotal with probably a close to zero average level of recognition.

La Belle France is mostly an English expression, notwithstanding that would be La belle France in French (unless it is a title), it is not commonly used here, especially with the stereotypical meaning you describe.

On the other hand, a pretty close well known French expression is :

Elle est belle, la France !

It is typically used in an ironical way by French people while complaining about some unpleasant fact about France.

It is well possible La Belle France was coined from Elle est belle, la France.