Barnard's exact test has one further advantage over Fisher's exact: Fisher's exact requires two fixed margins (e.g. the number of subjects in a treatment group and the number of subjects with a given adverse event), whereas in most places it is used only one of the margins is fixed (i.e. the number in a treatment group but not the number with a given adverse event).

The downside is that not too many software packages implement it. Specifically, SAS doesn't seem to implement it, so it doesn't get much use in the pharmaceutical industry. Having an implementation in R is a good start, so maybe more people will explore it and it will see more use.