Why not just use a proper secondary so that AXFR can handle the zone transfers automatically?
If for some reason you don’t want to use a secondary name server provider, you could also set up your own: Guide: How to setup NSD as a secondary nameserver for Mail-in-a-Box
I mean, theoretically, you could use a custom script to push the records to a secondary server, or even maintain them completely manually, but the whole process is much more error-prone that way. Also, every time you change a record or add a new one, you’d have to make sure that your script detects the changes and pushes them to the secondary server, or you’d have to make the changes manually.
A third option would be not to use DNS on Mail-in-a-Box at all, and instead use an external DNS provider, such as your registrar’s, which usually already provides several secondary DNS servers for free. Frankly, I would prefer this solution over a manual or semi-automated custom scripting solution any day 