Is this solder joint you speak of inside the pickup where I'd have to take the tape off? Or is it just on the bottom? Cause I remember seeing a solder joint on the bottom side.

It's the brass pins on the bottom. But it's the solder joint inside the coil I'm thinking is the problem. Heat the pins up with your iron so that the solder will reflow inside the coil. You can also take the tape off and get to the outside end of the coil winding and reflow it there.

Heat could certainly damage the pickup but, as I said, it doesn't seem like you have much to lose by trying. If it works, you won't have to replace it!