For the new unit I would apply fix #3 but was wondering which wire size is recommended to go around the connector?
I drew #3 showing relatively small wires. They run off to the board, which by itself doesn't draw much power. And I'm showing how to strip the red and black wires without cutting them, and then using the small wire to bind them together. What's not shown is a big lump of solder joining all that together. So the small wire is only holding the two larger wires in place while you solder them. It would then be the solder that carries the current, so a large enough lump would do that job.
If, instead, you're going to bridge the wires with another wire, then that wire needs to be the same gauge as the red and black wires (I don't know what that is off hand). But then you'd still need a smaller wire to run to the contacts on the board. That's why I drew it the way I did. The smaller wire is enough for the board, and the solder would handle the current for the SWG.
If you're going to do this to a new IpH controller, then you don't really need those little jumper wires leading to the board, because nothing has melted yet. The existing wires will do that job. To simplify all this, you can just cut all the red and black wires and splice four reds together, and then the four blacks together, using a couple wire nuts or a WAGO connector. I didn't draw it like that because those types of connections are still prone to corrosion, which is my theory for why the original connector burned out, after collecting enough corrosion to cause overheating. The soldering solves for that: it's a corrosion proof connection.
If you don't want to solder, and use a mechanical connector instead, you might consider some electrical grease to help protect the connection from corrosion. I got this off of Amazon. You can use this on all the pins of the exterior connectors, too. That's what I bought it for.
But you might consider waiting until after the warranty on the first unit expires. Pentair should replace it up until then. After the warranty is done, then do your spices. Otherwise, Pentair could deny the warranty on the second one (which should carry over from the first one).