I have had a 1l and a 2l and with my 1l I have not had an issue with head space. 1 cup of water is 235ml plus however much yeast you put it, liquid may bring you to around 500ml. Just shaking the flask periodically, in my eyes, won’t be an issue for you. Even if you used a stir plate I don’t think headspace will be an issue.
The last time I used my 1l I built up a strain over a week and did not have head space issues.