No, a betta can go LONGER than a week without eating and should be fine by the time you get back if you fed him before you left. Auto feeders can also cloud the water and it is strongly suggested you do not use it on a tank that size.
make sure u read the 6 day feeder thing instruction becuase on some of them they feed like 21 fish instead of 1.I learnt the hard way when i went vacation leaving my gouramis with the feeding block
As long as the water conditions are healthy and it gets a partial water change the day you go the Betta will be fine for 5-6 days without food to be honest. He'll be fine! I've left them 4 days and come home to entirely un-bothered by my absence.