Modify or override the Tiled Gallery template

In response to https://wordpress.org/support/topic/possible-to-modify-or-override-the-tiled-gallery-template?replies=2#post-6727884

I'd really like to be able to customize the Tiled Gallery template. The specific customizations I have in mind for my project are:
1. Replacing the "original image" that's hosted on the CDN with the actual original image on my server
2. Adding some markup to the gallery itself such as several wrapper elements to enable lazy loading using the [padding hack](http://andyshora.com/css-image-container-padding-hack.html) to maintain placeholder aspect ratios

I think these both may go a bit beyond just adding a few filters, but might lend themselves to a more efficient method of overriding the template altogether from a new, custom template file created within the theme directory.

Thoughts?

Start with the Tiled Gallery template and read the linked WordPress support discussion for the requested customization context. The work is complete when the project has a decided, documented way to replace the CDN original image and add gallery wrapper markup for lazy-loading placeholders.

Written by the indexing model from the issue text.