For reference, here is the full Pleasanton law:
It's pretty restrictive. But as Jay said, Pleasanton officials report that pretty much nobody ever files for a permit, despite there being many beekeepers in the city, and people have been anecdotally told "it doesn't really matter" by city workers.
However, if your neighbor is insisting on it, everyone will probably roll their eyes and go through the whole process, so you probably don't have much choice. And while a public hearing isn't the default, your neighbor could request one.
That said, I've also not heard of anyone being denied a permit if they follow the rules, and there's nothing in there that says your neighbors can simply object to it, as long as you're not causing a specific nuisance, defined as "Bees or hives shall be considered a public nuisance when the beekeeper's bees swarm, the bees abscond, or the beekeeping does not conform to this code, or hives are abandoned by the beekeeper".
As Jay said, contacting a friendly person working for the city is probably the best next step.
If it gets to the point of a public hearing, you should post that here: I think you'll find many people willing to come and support you in public.