You probably don't need a new core. Built out of heavy duty copper. Take it to a place that does core servicing and it should be done in an hour or so. They tank (clean) the core and the solder them. You have to be mindful of that core when replacing it and putting engine in. One bump and you screwed again. New is not always better.
You can install with engine in. It's just a little tighter to get in and get the nuts on. The fun stuff is taking the fender and hood off. You gotta break a few bolts to get them off.

Good luck.