ChargePoint Problem... anyone else?

A couple of general thoughts on soldering...

From the photos, this is a multi-layer circuit board with heavy copper traces and many via's going through the board. This makes sense because of the heavy current through the relay. This can make soldering very difficult because these same heavy traces act as a heat sink and draw heat away from the area you are trying to melt.

It sounds like you have the relay off (even if that required breaking it into pieces) which helps because you can work on removing / cleaning a single pin at a time.

It is not just the temperature setting of the iron, but also the size of the tip and how well you can make the thermal connection between the tip and the joint. Although this seems counter-intuitive, it sometimes helps to melt some new solder because the melted solder can pool and help to conduct the heat better. I assume you are trying to use a solder sucker and / or solder wick to remove solder? I tend to have better luck with a solder wick because you keep the iron in contact the whole time. Again sometimes melting some new solder onto the wick / joint can help to get more heat into the area.

Another thing that can help sometimes is to add extra heat to that area of the circuit board with a heat gun of maybe a hot plate (while trying to de-solder with your iron) if it is accessible enough. Of course this is usually awkward, requires 3 hands, and you have to be careful to not overheat or damage other parts on the board.

I would not worry about cranking up the iron. As long as the solder isn't melting It is not getting too hot

I agree that trying to somehow use wires is less than ideal.

As @techjmw mentioned, experienced solderers struggle with this kind of job.. It can be very frustrating.