Hi John. Welcome to the forum. There are a lot of really knowledgeable folks on this forum. I bet if you post a couple of fig and leaf pictures, somebody will be able to give you an idea about the type of fig you have. Who knows, it could be a unique variety.
There is a video somewhere on this forum that shows how to root a summer cutting by sticking it in a pot with one or two leaves still attached, covering it up with the top half of a 2-liter bottle, and presto.
Maybe you can practice rooting with your tree's new shoots this summer, since it doesn't produce that much fruit anyway. Those new cuttings, in pots, brought inside for the winter may reveal a gem.
I'm also in CT, and I bought a whole bunch of cuttings from Jon (pitangadiego on eBay). If you do a search for fig cuttings on eBay, you'll find that there are some people who are very reputable, and members of this forum.
check out some posts by Rafed, Harveyc, grasa, gorgi, desert dance, dieseler, driveway farmer. Sorry if I forgot to mention somebody obvious, everybody.
If I do well with my cuttings, I may be able to share some new plants with you this summer, I work all over Connecticut. .