Often, these sinks themselves are Corian, and completely bonded to the countertop. But you said yours is a Blanco, and the description of the black base material showing indicates something other than solid surface, so it should be removable.
Are there any screws involved in the brackets you mentioned?
If it is just sealed to the top with silicone, you could try slipping a stiff putty knife between the sink and counter, being careful not to ding up the corian too much.
You might contact a corian fabricator, as they would be familiar with the mounting technique which may have been used,.