The ideal setup is an SDR with DSDPlus software. This not only displays a log of all Radio ID's active in the last 24 hours, but also records each call to an MP3 file, named with the Radio ID. This way I am able to look up any call whether I missed it or not and listen for a badge number (sometimes I have to listen to it multiple times or listen to multiple calls to discern one)
However, since you're working with a standard radio, the pencil and paper method is probably your best bet.
As far as an app or database....The wiki listing is all there is. Your contribution would become the official listing for Greenwich.
Also, it's OK not to know the badge number, there are many records on the list where that is the case and we just mark it XXXX until such time that we are able to identify it. Even just the radio ID's would be a good start.