Try putting a fairly strong concentration of bleach in your tank and then flushing. Turn the water to the toilet off. Manually lift the flapper in the tank to let some of the water out. Dump a good amount of bleach. Then use the lever to flush and hold it down. Maybe a couple rounds of this.
Other than that i don't have a solution. You could try tiny pipe cleaners to clean the little holes in the rim.
When you live with dogs, you get very good at keeping the lid closed. That may make this issue worse. Agree with the bleach and that this is likely mildew, but would be more conservative with the quantity, maybe 1/4 cup or less. It can cause corrosion of any metal parts in the tank and start other colors, like green. It does need to sit for a while, at least 20-30 mins. before flushing, and you may need to do this periodically. On an odd note: I found this problem occurred after using one of the "sticky" bowl cleaners, the gel blobs that are intended to dissolve slowly. Apparently there was a form of mildew that had an appetite for the gel.