Thanks, but that’s not it; I had already checked that All is selected under Notifications.
In the meantime, the stock iOS Mail app now seems to be reliably providing push notifications using Activesync; I don’t know why it wasn’t working consistently before. I’ve settled on using that. It’s not as pretty as Spark, Edison and other third-party email apps, but it’s the only one that doesn’t store parts of your emails on their servers. Might as well keep using Gmail in that case. The reason why I got mail-in-a-box in the first place is to get away from hosted solutions with questionable privacy policies - and to learn about DNS and mail servers.