Instead of `exec`ing around, is it possible to run the backup command from another container in the same pod? Possibly by mounting enough volumes into both? Then you could just run the backup "cron" as part of the Gitlab pod, using cron itself, some lightweight alternative, or even just a shell script with a loop.
Looking at your YAML you are doing interesting things with "command" and "args" that I'm not sure work the right way. A form that has worked well for me is to use YAML's multiline strings to pass the script as a single argument to `sh -c`:
command: /bin/sh
args:
- -c
- |
POD=$(…)
kubectl exec …
Hope this helps!
/MR