As you have noticed, Toad DB2 installs and runs using its own DB2 Client\DB2 Copy.
The Toad installed DB2 Copy (i.e. TOADC47) is not eligible to be a default DB2 copy, but it is a DB2 copy none the less.
Spotlight on DB2 relies on a default DB2 Client\DB2 Copy. (That should be stated in the Spotlight release notes).
If you install another DB2 Client and make it the default, Spotlight will work using that DB2 Copy (i.e DB2COPY1).
You can however run Spotlight in your environment when you only have a Toad installed DB2 copy, by first launching Toad DB2 and then from within Toad Launching Spotlight on DB2. Once in Toad, you are connected to a DB2 LUW database you can use the Tools | Quest Tools | Spotlight menu item to launch Spotlight. Spotlight will inherit the Toad DB2 environment, (i.e. TOADC47) and be able to see and use any LUW databases you have cataloged in your Toad environment.
Hope this helps,