Try ${FUNCNAME[0]}
. This array contains the current call stack. To quote the man page:
FUNCNAME
An array variable containing the names of all shell functions
currently in the execution call stack. The element with index 0
is the name of any currently-executing shell function. The bot‐
tom-most element is "main". This variable exists only when a
shell function is executing. Assignments to FUNCNAME have no
effect and return an error status. If FUNCNAME is unset, it
loses its special properties, even if it is subsequently reset.