redirecting stderr to err function

Question

I want to trap errors with my err function, which writes to the stderr. But in this example I am getting two error lines. How can I remove the first error msg not generated by my err function? I want to prevent something like

_get_error_msg 2> /dev/null 

or

exec 2> /dev/null

err() {
  echo -e "[$(date +'%H:%M:%S')] \e[31mERROR\e[0m $*" >&1
}

test.sh

#!/bin/bash

set -Eeo pipefail

err() {
  echo -e "[$(date +'%H:%M:%S')] \e[31mERROR\e[0m $*" >&2
}

trap 'err line "${LINENO}": "${BASH_COMMAND}"' ERR

_get_error_msg

I am getting this output:

./test.sh: line 11: _get_error_msg: command not found
[04:18:36] ERROR line 11: _get_error_msg

How can I remove

./test.sh: line 11: _get_error_msg: command not found

Or pipe this line also to my err() function to get something like this

[04:18:36] ERROR line 11: _get_error_msg: command not found

just one line written to the stderr?

EDIT: Something like this:

#!/bin/bash

set -Eeo pipefail

err() {
  echo -e "[$(date +'%H:%M:%S')] \e[31mERROR\e[0m $*"
}

err_msg() {
  while IFS= read -r msg; do
    err "${msg}"
  done
}

exec 2> >(err_msg)
trap 'err line "${LINENO}": "${BASH_COMMAND}"' ERR


_get_error_msg