(1) On the desktop, clone the repo into a temp dir (where you will set all the branch labels the way you want them on the notebook).  Let's use a (bare) --mirror clone to make it non-tempting to fuss with a work directory, and to save some space, while also copying all refs in one swell foop:
desktop$ mkdir /tmp/for_notebook; cd /tmp/for_notebook  # or similar
desktop$ git clone --mirror /path/to/repo
Now that you have /tmp/for_notebook/repo.git (--bare and/or --mirror tends to add .git), set all the branch labels in this clone to match where they were on the notebook:
desktop$ cd repo.git      # i.e., /tmp/for_notebook/repo.git
desktop$ for refname in   # ok, now see below
Here's where you can either do things manually, or by script.  If there's a small number of branches you can just list them manually:
desktop$ for refname in fix/issue1 master; do
> git update-ref refs/heads/$refname refs/remotes/notebook/$refname
> done
If there are many, you can automate this with git for-each-ref, but it will give you long(ish, might as well just use full) names that require a little more shell scripting:
desktop$ for fullname in $(git for-each-ref \
> --format '%(refname)' refs/remotes/notebook/); do
> refname=${fullname#refs/remotes/notebook/}
> git update-ref refs/heads/$refname $fullname
> done
At this point git branch should give you just the branches you expect, but if there are extras, you can delete them with git branch -d.
(2) Now clone this to the notebook, as a new repo:
notebook$ git clone ssh://desktop.name//tmp/for_notebook/repo.git repo
This will no doubt set up the master branch already; you just need to add the other branches.  Re-updating master is harmless, so, same idea as before, except the remotes are now origin/* instead of for_notebook/*:
notebook$ for fullname in $(git for-each-ref \
> --format '%(refname)' refs/remotes/origin/); do
> refname=${fullname#refs/remotes/origin/}
> git update-ref refs/heads/$refname $fullname
> done
You probably want to tweak the config at this point, etc., so that you don't have desktop and /tmp/for_notebook/repo.git as origin.  (I usually do this by just editing .git/config directly.)  Compare with the notebook's original repo .git/config as needed.
(The old reflog is gone now, and any git stashes you had saved, as those were all local refs that did not get copied to desktop.)