I ended up resolving this issue by browsing the forums a little more and finding this comment by @Kevin.Herron that was key in fixing the label problems for me:
I cloned the gateway-webpage example (and added it as a module to my gateway to see how it works) and then immediately created a new project based on the client-designer-gateway-archetype. This then requires you to either set up a directory or package structure and for me, the issue was that the java and resources directories weren't 100% mirrored so some of the properties weren't being found.
Also, another thing that was tripping me up was that, for instance, in GatewayHook.java, we import the SettingsPage and SettingsRecord classes, but when you reference properties in that file (like nav.header and nav.settings.title - two of the labels that showed up incorrectly for me), they're actually coming from HomeConnect.properties, NOT HCSettingsRecord.properties.
If you set up everything correctly and you still get incorrect labels, you might need to do a gateway service restart. I restarted so many times (and repackaged the module) I don't know whether it would've worked the first time without a restart even if I had set everything up correctly.