There are a few false premises in this question that I'll have to address .

To start with, the Vatican does not have the power to suppress academic publications or this manuscript. Scholars publish articles and books that contradict the beliefs of Christians (and Jews, and Muslims, and...) all the time, without church interference. It also did not suppress the manuscript "back in the day" or at any point. The Decretum Gelasianum does pronounce it aprocryphal, along with a number of other texts, but none of those were "suppressed". There's nothing to suggest it was ever in wide circulation or widely read; no one had to track it down and burn it. We don't know of any communities that used it, in contrast to, say, the gospel of the Ebionites.

Second: the manuscript discovered in Turkey does not predate Muhammad. Though widely reported in the media to be 1,500 years old, this is based on a misunderstanding. Clearly legible in the published photographs, the manuscript itself says that it was written in 1500 AD. It is currently in the hands of the Ethnography Museum of Ankara (well outside of the reach of the Vatican), and they have not published anything further on it.

Third: we don't even know that it is the Gospel of Barnabas, since no scholar has formally identified it as such, few pictures of it have been released, and no one seems to have read it. It might just contain the Epistle of Barnabas, a wholly separate document that makes no mention of Muhammed and has nothing to do with Islamic theology.

Fourth: the odds are overwhelming that the manuscript is a fake. The Syriac in which its written is a mess, completely riddled with errors. It makes at least one terminological mistake (in referring to the Bible) that would be very unlikely for an actual Syriac monk to make. It has no provenance, so we can't trace it back any farther than 2000, and the museum holding it has not carried out any dating efforts on it (or haven't published them if they have). It's hard to tell whether it's a medieval fake or a modern fake, given all of that, but it's almost certainly not an authentic text of Barnabas.

Finally, on the gospel itself. After being mentioned in Decretum, it disappears from the historical record for about 1,100 years. That's a long silence, and a long time for something to exist with no record. It reappears in two manuscripts of the 16th and 18th centuries. Most scholars think it was written around 1600. Others like Jan Jootsen disagree with that date, but place it instead around the high Middle Ages (~14th century, usually).** About the Gospel of Barnabas generally - not the manuscript found in Turkey - plenty has been written.

*The arguments are summed up in Luis F. Bernabe's El texto morisco del Evangelio de San Bernabé.

** Jan Jootsen, "The Gospel of Barnabas and the Diatesseron", Harvard Theological Review 95:1 (2002): 73-96