I don't have chickens, still fighting blossom end rot too. Indoor with an HPS 400w, I ameliorated soil adding compost and whatever I can find. Seems easier to manage but still here.
Good trick to see in advance it will appear is checking the leafs. If they break making a little noise, feeling something hard cracking denotes there is too much food in the leaves, calcium is not necessarly missing, but mostly going to the leaves instead of the fruits. I've read in a few places and when I cut half of the leaves, it helps, but not definitely.
I'm starting to think I have to live with for some time at least until I find to manage it.
At least removing them helps the next ones, and removing the black end, the small green tomatoes are a consolation to cook with.