L.M.
I am so sorry for your loss.
As for how to handle the family...tell your cousin how happy you are for her and mean it. When the "labor stories" starts simply say "I know what you mean" or "I know how you feel".
If your aunt begins with something to the effect that you don't or couldn't because the cousins experience was so much worse, you can simply tell her that although each birth is different (as is each woman's response to it) your experience was very much like your cousins so you really do understand what she went through. Then simply decline to "discuss" it with her further.
My aunt and mother have fueded since they were kids and they haven't outgrown it yet. When I had my son (almost 17 years ago), my aunt called to congratulate me (we always got along ok although I believe her to be off her rocker) she proceeds to start in about my mom. I actually had to tell her I had company...truth was, I have never been in the middle of their feuding and refuse to start now and it was really the last thing I wanted to discuss at such a joyous time. The point to that is that regardless of the distance between you and your cousin, she will probably be more than happy to keep things civil right now. Because of your recent miscarriage, she will probably feel ackward "gushing" about her new bundle of joy. Good luck!