The inflating during the day and deflating at night is normal. Don't sweat that. The dead part will over grow in time.

When my 135 crashed last year, my bubble that I had had for 7 years lost one whole side. I thought I had lost the enitre coral. I put it in my hospital tank on the faint hope it would somehow survive and a couple of months ago I moved it back to the main display tank. Other than it is about a third smaller than before, it has completely healed (around 4 months time).

One thing I do when I feed mine, is I mash the shrimp up between my fingers (after removing the carpiece) to soften it up before dropping it onto the tentacles. If you don't, then it may take the food in, but sometimes, usually when you are not looking, it will spit it back out and it is covered in some sort of mucus and stinks a lot. That stuff will then a foul a tank very quickly. So you need to keep an eye on it and make sure that it digests all of the food you feed it. I also do this same thing with my BTA as they will sometimes expel the shrimp if it is whole as well.