Installing FEniCS from source without admin privileges on a cluster would be quite difficult.
You cannot run a docker image on a shared cluster, because docker images provide the user with root access inside the image and this is not acceptable in a shared environment like a HPC cluster. Singularity was built exactly for this purpose. Once you have singularity installed on the cluster, you can use the command
singularity pull --name fenics.sif docker://quay.io/fenicsproject/stable:latest
to pull from docker and build a singularity image. You can use this to run your code, just like with docker images, using the command
singularity exec fenics.sif python3 my_code.py
To build fenicsx, you would have to pull the appropriate image from docker.
singularity pull --name fenicsx.sif docker://dolfinx/dolfinx
See this link for more information.