... But I think you can put some pool chlorine in it. I don't know the ratio because I'm gonna assume the birds, et al., will be drinking it. Our public water supply is chlorinated but not enough for your purposes.
If you don't get any help here, search the WEB for Bird Links and call or e-mail an 'expert' of sorts. You may find out that there isn't anything you can do. It's always beneficial to 'refresh' the fountain with new water on a regular basis anyways to keep it clean. Soooo, you may find out that what you're doing is all that can be done.