How thick is the briar where the hole is? I once repaired a Grabow author shaped pipe with a hole in the bottom of the bowl. Hole was quite a bit bigger than the one in yours. I used a three part repair which consisted of pipe mud,JB weld and wood filler.
After cleaning the bottom of the bowl thoroughly,I put a thin layer of JBW over the hole being careful not to push it all the way though. When it set up I put a layer of pipe mud over the JBW. On the outside I filled the hole with wood putty. When it dried used an Exacto knife and small files to duplicate the rustication,stained it and shellaced (just the repair) it to match the lacquer. It smoked fine, no weird tastes from the JBW. JBW is good for 600 degrees.
Another option would be a briar plug.