One potential technique is just to clone a single branch. You can then pull in more later. Do git clone [url_of_remote] --branch [branch_name] --single-branch.

Large repositories seem to be a major weakness with git. You can read about that at http://www.sitepoint.com/managing-huge-repositories-with-git/. This article mentions a git extension called git-annex that can help with large files. Check it out at https://git-annex.branchable.com/. It helps by allowing git to manage files without checking the files into git. Disclaimer, I've never tried it myself.

Some of the solutions at How do I clone a large Git repository on an unreliable connection? also may help.

EDIT: Since you just want the files you may be able to try git archive. You'd use syntax something like

git archive --remote=ssh://git@bitbucket.org/username/reponame.git --format=tar --output="file.tar" master
I tried to test on a repo at my AWS Codecommit account but it doesn't seem to allow it. Someone on BitBucket may be able to test. Note that on Windows you'd want to use zip rather than tar, and this all has to be done over an ssh connection not https.

Read more about git archive at http://git-scm.com/docs/git-archive