Is there a firewall on an iPad on iOS 10, and is it configurable?
There is no firewall on iOS and it does not block any ports that I am aware of.

I'd guess your issue has more to do with the fact that Bookman hasn't been updated in over three years and some change in iOS 10 finally broke it. I'm actually shocked it works at all.

It sounds like you can use iTunes file sharing as an alternative, presuming that a recent iTunes update hasn't caused that to stop working with this particular app.