It should be possible. Although probably very complicated. Have a look at https://distrobox.it/. It allows you to tightly integrate containers into your desktop, including accelerated graphics, some devices, your homedir, etc. It can even automatically install desktop shortcuts. (You can disable the integrations of course) Even tho it uses Podman instead of docker, AFAIK it should be 1:1 compatible with docker for your usecase.
You can even have all nixpkgs on a different distro through the nix package manager