No I don't believe muriatic acid would harm the toilet, but you don't need the acid to be that strong.

Toilet bowl cleaners come in two flavours; phosphoric acid and hydrochloric acid.

Any place listed under Janitorial Equipment & Supplies will sell both types.

Generally, it's better to use the phosphoric acid if you can because it's easier on the environment, but the hydrochloric acid at the 10 to 15 percent concentration typically found in toilet bowl cleaners WILL NOT harm porcelain. I looked at the picture of your toilet, and I don't see anything there that either kind of cleaner wouldn't dissolve. Hydrochloric acid based toilet bowl cleaners are very much stronger than phosphoric acid toilet bowl cleaners, so you might want to remove most of that brown stain with phosphoric acid, and then clean up the more difficult stuff with hydrochloric acid. If you have muriatic acid, I'd use it.

I use hydrochloric acid toilet bowl cleaners all the time on porcelain because I often clean my aluminum window screens in bathtubs, and the aluminum leaves marks on the ceramic tiling. Hydrochloric acid is the only thing I know of that will remove those marks. And, of course, I use hydrochloric acid to clean toilets as well.