Hi Todd Hinds,
Good Day. I hope everything's well.
In regards with your installation issue, you can try to sign-in through Microsoft Account inside Office (word, excel, outlook, etc.). Devices are automatically registered when you sign in with MS account. On Desktop PC, go to official installer https://www.office.com > Sign-in with the same Microsoft 365 account you used on your 2 devices (laptops) < once Signed-in, click Install Office > Microsoft 365 apps. Run the installer and complete setup. This automatically registers the desktop to your Microsoft 365 subscription - no serial number required.
Second, Check also if your current subscription is active. Go to https://account.microsoft.com/services and check if it is listed as "Active".
Note: recommended to uninstall any old or preinstalled Office versions. Uninstall all entries named Office or Microsoft 365. Reboot your PC and reinstall Office using the first procedure.
I hope this helps.
Regards,
Jhun