How can I make -e catch an error in a script within a script?

This might not be your case, but did you call banana.sh in a specific context where -e is being ignored?

In the Bash reference manual you can read a detailed explanation about such a context, with the important info as follows:

If a compound command or shell function executes in a context where -e is being ignored, none of the commands executed within the compound command or function body will be affected by the -e setting, even if -e is set and a command returns a failure status. If a compound command or shell function sets -e while executing in a context where -e is ignored, that setting will not have any effect until the compound command or the command containing the function call completes.

This means if you have the following apple.sh:

set -e

func() {
        echo '...some commands #X...'
        false # Suppose ./banana.sh fails here
        echo '...some commands #Y...'
        true
}

if func; then
        echo 'func() returns success'
else
        echo 'func() returns failure'
fi

You'll get the following output:

...some commands #X...
...some commands #Y...
func() returns success

Not only bash, but other shells including dash and zsh will give the same result.

To answer your question, there's apparently no way to change the behavior at least on bash.