I would give the CLR product a shot. Followed by a good rinding to remove the chemical. This is sold in any home supply store. I got mine at Walmart

Recalling high school chemistry and the chapters about acids, alkalies and potential of hydrogen (PH).

I'd clean it with plain white vinegar and water. About a 1 to 1 mix. Let the liquid set for an hour or so, then scrub with a stiff brush or 3M scrubber.

You could add a cup of vinegar to the fresh water when you fill the bird bath. The acidity of the vinegar will lower the PH of the water and help keep the alkaline mineral deposits down. The birds will never notice.

BTW I have a ground level bird bath and my problem is an algae like slime. I normally just hit it with the hose to blow out all the "solids" and refill with fresh water. However, I clean it once or twice a summer with a little beach in water and then let it set out in the full sun to dry for a day. But, it gets used a lot and it seems to slime up very fast.

white vinegar, definitely. The less chemicals, the better. I also use vinegar to kill weeds (spray it on the plant or pour a capful on the roots)