In case anyone comes across this question in the future, I found a plugin that does everything I want.
The plugin is Multisite Post Duplicator which is a free plugin available in the WordPress plugin directory. Below are some of the plugin features.
Features:
- Copies all custom fields
- Copies all related post meta
- Includes any custom post type on your network as long as the post type exists in your destination site
- Automatically copy your post/page/custom post type from one site to another from within your workflow
- Copies any featured image (Can be turned on or off in Settings)
- Copies all image media within post content to the new site’s media library for exclusive use in the destination site (Can be turned on or off in Settings)
- Copies associated tags (Can be turned on or off in Settings)
- Batch Duplication
- Settings page to customise the default behaviour
- Restrict functionality to only certain sites on your network
- Clean and friendly User Interface
- Select what status you want your new copy of post to be i.e Published, Draft etc
- Specify a prefix for the new post to avoid confusion
- Works with Contact Form 7
- Works with Advanced Custom Fields