Are you saying you paid $100 for the Win11 Pro license? If so, you have the upgrade license, not the full license for Pro.
An upgrade license can ONLY work if it is used to upgrade an "already" activated Home edition. If the new computer doesn't have any license at all, the upgrade license alone doesn't work. Your options is:
Purchase Home license, install as Home, then do the following:
You can go to Settings > System > Activation > Change product key
Disconnect internet, and disable any third party antivirus, key in the generic key instead: VK7JG-NPHTM-C97JM-9MPGT-3V66T
That should get you to Pro. Once restarted, you can connect back to the internet, sign into your Microsoft account. See if it is activated. If not, go to Settings > System > Activation, run the troubleshooter, choose the hardware change option. When it asks you where it is installed on right now, you should choose the old device name. Then click on Activate button. And on the next screen when it asks you where to transfer it to, choose your new device name.