A lottery is a lottery - there can't be special preferences that deprive some kids of the opportunity just because they may have a stay-at-home parent or a sibling. That's not fair to the other families.
The problem you face with half-day kindergarten is the same one faced by parents with kids in half-day preschool - what to do with the other half of the day. Those half days, the days the programs are closed for teacher in-service days, the school vacations, and the snow days are all challenges for working parents or those with multiple children. Frustrating but not unique.
Our district has an early drop-off program that accommodates working parents or those who are driving 2 kids to 2 different schools. It's paid for by the parents who use it, and it has a certified teacher in charge of it (but not a teacher involved in the regular classrooms).
Sometimes other organizations (the Y, the JCC, etc.) offer half-day programs for kids doing half-day kindergarten. Our JCC did one for years, sometimes with bus service (if on a regular bus route from the elementary schools) or with their own van. Kids could be enrolled 2, 3 or 5 days. Of course, in home care with a nanny is an option used by different families. Some offset that cost by having their nanny watch another child as well (for a fee - no different from regular babysitting).