I originally made this using a PNG as a template, I inverted the colors, cleaned it up, etc. in GIMP. 

Afterwards, I deleted some text and elements, added in my own text and elements, aligned everything, etc. in Inkscape.

Here's the part I need help with
I want to make most of the white background (especially the parts outside the outer circle) and make that transparent. I tried following the steps here, but I don't think it worked because of the way objects are grouped
https://inkscape.org/forums/questions/making-a-specific-color-transparent/

Is there an option that will let me add transparency and keep my design without having to remake all of the text and graphics from scratch?

I'd probably use clipping:

http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Clip.html

Thanks so much @TylerDurden . I'm going to try this later!

You could try this, if you want transparency inside the logo.

Open the group.
Select the bitmap image.
[Object > Fill and Stroke] or [shift+ctrl+f].
At the bottom of the Fill and Stroke dialog, set [Blend mode: Darken]

Thank you @ Paddy_CAD . I use the clipping method and I am happy with the results, but I was curious how to achieve transparency inside as well.

Here is a peek at what I'm working on (all the text details are tentative/placeholder, but the logo looks natural on top of the graphic!)