Thanks to @sunny7777777 for pointing to the workaround. For anyone else using Cordova 3.1 or higher, you can disable websecurity using this command:
<preference name="websecurity" value="disable" gap:platform="blackberry10" />
@candlejack added in the comments: Set to disable to override web
security settings, allowing access to remote content from unknown
sources. This preference is intended as a development convenience
only, so remove it before packaging your app for distribution. For the
released app, all URIs should be known and whitelisted using the
element, described in the Domain Whitelist Guide.
I am now convinced that this workaround should only be used during development and not in production. This is true for all platforms, not just BB. So be careful to remove this before publishing your app!