There's a reason why they call it the "fixed" cup.

While Italian bottom brackets undo anti-clockwise, I doubt that applies in your case. A quick check is to measure the width of the frame's bottom bracket shell. 68mm = English, 70mm = Italian.

There's a couple of tricks to removing seized fixed cups.
The first is holding the removal tool in place. A long bolt through the center or a big C-clamp, either just slightly loose, is a major help.
The second is leverage. If you're working with a 12" Crescent wrench, you probably need more leverage.

If you have access to a sturdy bench vise try clamping your removal tool vertically in the vise. Get someone to help you thread the frame onto the tool and hold it flat. Think for a minute about which way you need to rotate the frame and give it a spin. That trick has never failed me.

Oh - and you can leave the fixed cup in place, clean it out, and replace the bearings from the opposite side. If the left side cup isn't pitted, just be sure to examine the fixed cup as well as you can with a flash light.