Hi Christian, thank you for posting here in the Microsoft community, to update the Microsoft Edge Update component to version 1.3.203.13 on Windows Server 2019, especially when it's stuck at 1.3.201.11, you can try this step.
Check via Edge Components Page, Open Edge and go to: edge://components then look for Microsoft Edge Update and click Check for update.
Also, you can use Manual Installer, simply Download latest MicrosoftEdgeUpdateSetup.exe from Microsoft Edge Enterprise site, https://www.microsoft.com/en-us/edge/business/download then Run the installer manually on affected machines.
If you're automating this across multiple servers, consider using PowerShell with the Edge Update API to fetch the latest version dynamically.
I hope this helps you out.
Chard