Andrew, I believe they are called 'finger plates'. Most good joinery or ironmongers/tool shops will keep them. However they are usually long and oblong.
You can get 'kickplates' for the bottom of the door and these are often made to order due to the multitude of sizes possible. You could order postcard size kickplates? They would come ready drilled and countersunk and no doubt you could ask them for the hole in the middle too.
No idea how long this type of thing takes though, is this job ready to go?
How about approaching a local 'metal worker'? He could knock up something surely?