Unable to authenticate against private registry for docker pull codeclimate engines

There are multiple docker in docker levels occuring when running codequality in a gitlab pipeline and it is currently impossible to mount a config.json deep enough for authentication to docker pull codeclimate engines:

VM - runs gitlab-runner
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`docker:stable` - docker run for each gitlab pipeline steps
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`codequality:0.85.26` - docker run at the codequality step in the pipeline via `run.sh`: https://gitlab.com/gitlab-org/ci-cd/codequality/-/blob/master/run.sh
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`codeclimate/codeclimate:0.85.26` - docker runs engines:install etc. **<- we are failing here**

I can mount --volume /root/.docker/config.json:/root/.docker/config.json:ro with the --volume /var/run/docker.sock:/var/run/docker.sock for docker:stable and codequality:0.85.26 levels this allows me to pull level 4: codeclimate/codeclimate:0.85.26 from a private repo.

However, I cannot mount config.json into the docker run codeclimate/codeclimate:0.85.26 engines:install step within run.sh of codequality:0.85.26. This will attempt to pull myregistry.com/codeclimate/codeclimate-structure and will fail.

...
$ docker run --rm \ # collapsed multi-line command
Error response from daemon: pull access denied for myregistry.com/codeclimate/codeclimate-structure, repository does not exist or may require 'docker login': denied: requested access to the resource is denied
error: (CC::CLI::Engines::Install::ImagePullFailure) unable to pull image myregistry.com/codeclimate/codeclimate-structure

I cannot add a docker login step into the run.sh of codequality:0.85.26 and I cannot mount config.json so how can we authenticate against a private registry to docker pull codeclimate engine images?