Do not use a pointed punch as it will spread the end of the pin and you will not get it out. (A stainless fitting shop will have same thickness stainless rod and you can buy a foot or so.)
If the pins are really stuck with corrosion and space is tight, then use a dremel with a mini metal cutting wheel. There is enough room to get the wheel between the hinges. The hinge pins are soft metal and cut easy. Then use a piece of the cut pin and a pair of pliers to press/push the rest of the pins from the hinges. Once you get a little out you can then twist the old pin out using the pliers.
The stainless shop will also have 316 bits to make an as good thread, washer and butterfly nut. I preferred 316 to the chrome over brass OEM part.