Cannot update to 13.1

Shortly after posting this, restarting the Mac and quitting the softwareupdate process from Activity Monitor, looks like searching for macOS in the App Store and then selecting the Get button finally forced the 13.1 update to kick in.

Oddly enough, this issue continues when trying to update from 13.2 to 13.2.1 from System Settings > General > Software Update. Even after running below command from Terminal as previously suggested by P. Phillips:

sudo softwareupdate -i -a -R

If however I go through the App Store as I did the last time, 13.2.1 does pop-up to install, but it seems to be downloading the full 12GB Ventura installer rather than just the update.

So this time I went even further and deleted com.apple.AppStore.plist files from ~\Library\Caches, ~\Library\Preferences and /private/var/folders. Restarted my Mac and tried the Software Update again. Still no luck. And so, yes I could keep downloading the entire macOS Ventura installer from the App Store to update my Mac, but feel like this is probably not the ideal long term solution. I'd appreciate if anyone has further troubleshooting steps for me to try and in the meantime I'll be staying on 13.2.

Thanks for your guidance. That seems to have indeed downloaded the update and placed Install macOS Ventura in the Applications folder. I imagine this is the same package that would have been downloaded from the App Store. It's just unusual that Software Update wouldn't pick this up.