Hello,
Situation with running Docker containers inside the official Docker-based LAVA setup (https://git.lavasoftware.org/lava/pkg/docker-compose) was already discussed at least once recently.
LAVA starts to offer more and more convenient usages of Docker containers to extend its functionality (e.g. https://connect.linaro.org/resources/ltd20/ltd20-304/), and people start to leverage it. For docker-compose based setup, that effectively means running Docker containers inside the docker-compose, and that currently doesn't work.
Another reason for raising this matter again are the apparent plans to migrate components of the production LAVA setup as run by Linaro (https://validation.linaro.org/) to Docker-based setup. The concern is that if Docker-in-Docker scenario is not explicitly accounted and planned for, then there will be a regression due to this switchover.
So, I posted https://git.lavasoftware.org/lava/pkg/docker-compose/-/issues/7 with more detailed discussion, and would be ready to follow up with a patch (we just need to decide what kind of patch, please the proposed options there).