Do the Bellaigue Benedictines still exist?

They surely must. I have not heard anything to the contrary.
In 2019 I recall that the SSPX ceased promoting them. They were removed from SSPX websites. This was soon after the agreement between the SSPX and modernist Rome on marriages, but I don't know if that was the cause.

This traditional monastery was founded in 2000 by four monks from the then Dom Tomas Aquinas's monastery of the Holy Cross in Brazil:
https://fr-m-wikipedia-org.translate.goog/wiki/Abbaye_Notre-Dame_de_Bellaigue?_x_tr_sl=fr&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=sc
In October 2000 , four Benedictine monks from the monastery of the Holy Cross of Brazil settled in the abbey. They rebuilt the vaults of the abbey church, which allowed the return of Catholic worship.March 9, 2008, Father Ange Araújo Ferreira da Costa, first prior of the new foundation, died at the age of forty-three, in the eighteenth year of his monastic profession and the fourteenth of his priesthood. His remains rest in the abbey.

According to this SSPX article from 2017 reporting their founding of a new monastery in Germany, they then counted around 40 monks:
https://fsspx.asia/en/news/inauguration-new-benedictine-monastery-17758
For the time being, Reichenstein Abbey remains dependent on the Benedictine monastery of Bellaigue (near Clermont-Ferrand), which was founded in 1998 in one of the oldest abbey complexes in Burgundy and today counts around 40 monks.

What a shame all of these "resistance-minded" monasteries were "too prudent" to stand up with the Resistance when it mattered:
https://www.facebook.com/mariancorpsofst.piusx/posts/the-benedictines-of-bellaigue-no-more-than-the-capuchins-are-known-to-be-resista/756732458122774/

There is a new development:

As the SSPX broke off ties with the traditional Benedictines in Bellaigue in 2019, it also broke off ties with Bellaigue's filial monastery in Germany, Reichenstein Monastery.

The latest development is that Reichenstein Monastery in Germany has separated from the "resistance" and is now back in touch with the SSPX.

A German-language article on the current situation:
https://fsspx.de/de/news/ein-kloster-knuepft-wieder-priesterbruderschaft-st-pius-x-50372

We learned from Fr Chazal just recently that Bishop Morgon visited Bellaigue and offered a pontifical Mass there, thus confirming what Bishop Williamson had said in a recent Eleison Comments that they are now part of the Resistance. The German monastery that they had founded clearly does not share their clarity of vision regarding the crisis in the Church and the SSPX - evidently they are happy to live in Huonderland:

Communiqué of the Monastery of the Immaculate Heart of Mary Reichenstein (Germany)
Due to the serious internal problems in the monastery of Notre-Dame de Bellaigue and the fact that the “resistance” was used for the granting of orders, we distance ourselves from the authority of Dom Placide (Prior of Notre-Dame de Bellaigue, our Motherhouse). Our conscience no longer allows us to depend on him.
We want to continue our monastic life in fidelity to the Benedictine tradition and to the tradition of the Catholic Church, alongside the Priestly Fraternity of St. Pius X and in friendship with her.
Reichenstein Monastery, the 2nd February 2025,
at the festival Mariä Lichtmess
P. Bernhard O.S.B., Prior
P. Benoît O.S.B.