I’m attempting to automate moving between different time zones while I travel and was curious whether someone knew how to update the “Additional Clocks” setting by using the Windows API/SDK. I’ll be using .NET and C# but would definitely be happy with the raw API calls as well. Thanks!

That’s not how it works. The winapi would grow gargantuan if there was an api for every single little tweak dialog like this one. Instead, the dialog changes registry keys, keys that are later read back by whatever code displays the clock.
You can find them by running SysInternals’ ProcMon. Start it just before you click the Apply button, stop it right after that. You’ll have little trouble finding the HCKU\Control Panel\DateTime\AdditionalClocks being used in the trace. Mapping settings in the dialog to registry values is pretty straight forward.
Hacking registry keys that belong to Windows is pretty iffy. But you’ll get away with it as long as you don’t hope this still works in the next version in Windows. Actually changing them might require a logoff and logon to make the changes effective, I didn’t try that.