You are right to say there's no medical data on chest belt & GERD.
My opinion is that wearing a belt around your chest helps with one of the most common types of hiatal hernia, called sliding hiatal hernia, which has a main feature of the hernia sliding in and out of the lower chest.
In such a case(sliding hiatal hernia) the diaphragmatic muscles involved in the opening and closing of the lower esophagical sphincter can become dysfunctional, causing the GERD symptoms.
By tightening the chest belt, you may be arresting the sliding movement of the hernia, causing a temporary reversal of the dysfunction, to the point of significantly alleviating the GERD symptoms.
And, yes, it is vital that your breathing remains normal, while wearing the chest belt.
As a long term non-surgical solution, please look at the possibility of Chiropractic Hiatal Hernia Reversal, if you are a suitable candidate.
Non-invasive and very simple.
I hope this helps,
Niko