Can I make a "Create Gallery" tab with different output than [gallery] ?

I’m wanting to create a tab similar to the “Create Gallery” one which appears after clicking the ‘Add Media’ button.

Here’s something of an example.

Something that would let a user select one (or more) files of any type, then insert a shortcode such as [media-gallery ids=”3,5,8,16,24″] into the TinyMCE editor.

I’ve spent roughly 6 hours this morning searching some sort of solution and have yet to find any good leads. ??

Any thoughts or suggestions are very greatly appreciated!

(X-post from StackXchange: https://wordpress.stackexchange.com/questions/162246/how-to-make-a-customized-create-gallery-tab-in-the-add-media-window )