If anyhow possible I think the Avatar should be copied to and served by jenkins. This assumes that you can access the actual image file through the gitlab api to copy the file over.
If this is not possible, then I don't have a strong opinion about how to solve this. I just think jenkins should never ever show a broken image (to be exact, the browser shows no image at all when the user isn't currently logged into gitlab).
Making this configurable as a trait would be just a workaround. I still wish to see the actual avatars of the gitlab projects, but I think there should be a way to archive that even if the user is currently logged out from gitlab.
If course, this is only a minor issue and not a deal breaker at all. You shouldn't make it a priority to fix this for the next release
(I think it is a much bigger deal that all branches and tags are automatically built when first importing a project into jenkins, even when using "skip initial build" of the "Build strategy plugin". I will try to validate this bug tomorrow before issuing a bug report.)