Not without nuances. That covers the bases for license compliance.. but implementation goes, I will unpack further here -
You cannot use the home license directly on a pro edition due to edition mismatch. You can only activate it on a home edition. But there's no downgrade path from Pro to Home, it will require a clean install to go back to Home, then apply your Home license. Then do the following to upgrade to pro: 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.
Another option is, once you have home license purchase record/proof, contact Microsoft support, see if they would help you activate without going through that big loop above.
To get help, open the "Get Help" app on your computer (just search for it in the Start menu). or go to https://support.microsoft.com/contactus, Type "Billing" and press Enter. On the next screen, click "Contact support" at the bottom of the page, then choose "Windows" > "Manage my subscription". You’ll then see the option to chat with a Microsoft support agent.