How to force Backup Assistant to sync in airplane mode with WiFi on
Definitely contact the dev about that, like Izzy suggests.

You could simply not use Airplane mode (you're not really saving battery life if you're still enabling WiFi and other radio connections like Bluetooth and GPS) and just disable mobile data on your device. 
This article indicates that you can easily disable Data by going to Settings > Battery &amp; data manager > Data Delivery > and un-tick the Data enabled option. This might vary across devices and carriers.

That should allow you to use your device without a mobile carrier network connection, saving battery by not searching for the network, and allow the app to work just fine.