Well, I've been thinking... 21st century lyrical anachronisms aside, wouldn't they both fit nicely into Hairspray??!! I'm sure Tony-winners Scott Wittman and Marc Shaiman could fix those 21st century anachronisms - Tracy Turnblad wouldn't know what "photoshopping" is, would she?
Here's how "All About That Bass" might work in Hairspray: In a final televised plea for integration and for Miss Hairspray votes, Tracy sings the song to Corny Collins and "the Nicest Kids in Town." Eventually, Tracy, Penny and Link (plus a few of the Council Members who secretly agree) go into a full blown dance with it, and the phones ring off the hook. At the same time, we can see The Dyn-o-Mites, Motormouth Maybelle and all the kids at her record shop watching this on TV, and Edna and Wilbur are home watching it, too. They all join in until it becomes a huge production number.
The commute has been hell lately, so I've had way too much time to think about this! I can think of at least two more scenarios for these songs in that show. Ah, well... gotta get to work!