That is definitely interesting. Sure enough MonthMixin includes get_next_month/get_prev_month methods, and DayMixin includes get_next_day/get_prev_day methods. However, both YearMixin and WeekMixin have no functional equivalent in their definitions. Seems like a bit of an oversight on the Django team's part.
I think your best bet is to subclass either WeekArchiveView or BaseWeekArchiveView (if you may eventually want to change up the response format and don't want to have to re-implement your methods) and add your own get_next_week/get_prev_week methods. Then have your view inherit from your subclass instead. A simple modification of DayMixins methods should be sufficient.
def get_next_week(self, date):
"""
Get the next valid week.
"""
next = date + datetime.timedelta(days=7)
return _get_next_prev_month(self, next, is_previous=False, use_first_day=False)
def get_previous_week(self, date):
"""
Get the previous valid week.
"""
prev = date - datetime.timedelta(days=7)
return _get_next_prev_month(self, prev, is_previous=True, use_first_day=False)