Thanks for your post.
Licensing recognizes hardware, so even if you name it the same it will still think you're on a different machine.
You can perform the following though:
1) Download the trial version to Server B
2) Take a backup(s) on licensed Server A
3) Copy backup(s) to Server B
4) Perform restores with trial copy for 14 days
5) During that 14 days, finish up with Server A
6) Prior to decommissioning Server A, right click on the server through the SQL Backup GUI and select to deactivate
7) Uninstall SQL Backup from Server A
8) Decommission Server A
9) Use key from Server A which has been deactivated to activate Server B
Red Gate Software Ltd
+44 (0)870 160 0037 ext. 8569
1 866 RED GATE ext. 8569