Turns out, you can actually use your Apple Watch like an iPod. There are a few steps you have to go through to get it going, but that's not terribly different than setting up a brand new iPod, so we're not mad at it.
So, how do you do it? First, you need to transfer your songs: Apple Watch App (on phone) --> Music App
--> Select a Playlist Limit --> Sync Playlist. Note that it does need to be charging to sync the music, so you probably don't want to do it as you're heading out the door. In fact, syncing while you sleep might be your best bet.
Once you're all synced up, pair your headphones or speaker and you're 90% of the way done. Now, just go to the music app on the Apple Watch, hold your finger down on the screen and you'll get two options: "Source" and "Now Playing." Choose "Source," then "Apple Watch" and you're good to go!