@sobi3ch consider this content in .git/config file contents:
[core]
repositoryformatversion = 0
filemode = true
bare = false
logallrefupdates = true
[remote "origin"]
url = git@........git
fetch = +refs/heads/*:refs/remotes/origin/*
push = HEAD
[remote "github"]
url = git@........git
fetch = +refs/heads/*:refs/remotes/github/*
[remote "newrepo"]
url = git@........git
fetch = +refs/heads/*:refs/remotes/github/*
Now since you have your own repo "newrepo" to work with but at times you feel like pulling contents out of the original repo... you can do all the commands specifying where to pull differences from.
So the steps would be, clone that single branch.
Push to your new repo only this branch
(git push -u newrepo branchname
) for example.
Hope this answers essentially the question.
By the way, newrepo will only have the cloned content from original repo, and the changes you provide, regardless where they come from.
I'd also protect myself from using common "origin", by removing it.
I wouldn't use git remote update, that will act against all remotes.