I think that you’re looking for the pipefail option. From the bash man page:
pipefail
If set, the return value of a pipeline is the value of the last (rightmost)
command to exit with a non-zero status, or zero if all commands in the
pipeline exit successfully. This option is disabled by default.
So if you start your wrapper script with
#!/bin/bash
set -e
set -o pipefail
Then the wrapper will exit when any error occurs (set -e) and will set the status of the pipeline in the way that you want.