What is the best way to deploy a compiled language project with docker and jenkins?
I do not know how how about Jenkins, but GitLab CI has feature called artifacts which allow you to


explitely share some files/directories between builds
download them from GitLab Pipeline view


Additional pro is that allows you to make deployments manual actions.

Also if you are building fat JARs you do not need Docker for your deployments at all.

So the best solution for you is to have separate task for building release and separate for deployment (which can be manual job or not).