Discussing finances with advisor: Am I being selfish?
What you are getting paid when you are not working for your university should not matter. Taking it to a (albeit artificial and ridiculous) extreme, if a top programmer in industry, with a salary of 100k$/year joins a doctoral program in June, would you consider that his salary from January to May is enough so you can not pay him for a couple of years? In my view (and some jurisdictions too), working for free is very close to slavery, and should be avoided when possible.

So, absolutely, you can go ahead and ask her if she will fund you. Your situation is good in the way that if she says no you will have a backup.

With this said, I can understand if she refuses. If it is a single, non continued project, it is not as efficient to hire a person for one month as for three, and if you are not going to do it, she may use the funds to hire someone else for the full time. If you are going to continue doing your usual work during that month, there is not so much difference between being there one or three months, and your case is stronger.

As an extra, during your fellowship, you will probably learn a bunch of new things, so you are an extra asset for your group. You can also offer to give a few seminars on what you have learned in partial exchange for the funding.