I have never noticed those noise dampening pads for sinks for sale anywhere. I would think they are available somewhere. But what you want is something that is dense enough to absorb the noise and will permanently stick to the bottom of the sink. If it were me, I would use a patch of "Stick-it-n-forget-it" tape, sold through the Books & Parts link above. The tape is close to an 1/8" thick, made of very soft and sticky rubber, with a white polyvinyl skin. When I want this stuff to really stick, I will press it into place with a metal laminate roller. A roll of coins or anything shaped like that and hard should work fine. The more pressure applied, the better the stuff sticks.
Another idea is to get sound deadening film from an auto sound shop. They use this stuff on the doors and chassis to reduce noise.
Last but least is down and dirty. The black roof mastic would not be pretty, but it would certainly work and is real cheap. I imagine that you could use a white elastomeric, but I would imagine it would take many coats to get it thick enough to work.
Just some ideas.
Today is PERFECT!
All information and advice given is for entertainment and informational purposes only. The person doing the work is solely responsible to insure that their work complies with their local building code and OSHA safety regulations.