Yep, as you can probably gather from the title of this post, it finally works. I rebuilt the whole thing only to find out that it still didn't work. I had a high degree of confidence that my hardware was done right (both times), so it had to be the code. Sure enough, after staring at it for hours, I found it. There was a single character in the string I was sending to the Skywire cellular modem that shouldn't have been there. Stupid plus sign. As soon as I removed the offending character, NAGBox starting sending text messages like a champ.
Now I just need to make 3 more! And now that I know what size the board & components will be, I can design and print the cases for them. Looks like I'm not going to get these done in time for Christmas after all. That's a bummer. Oh well, it'll be the best post-Christmas present my family members ever got.