L.R.
I am so sorry for you, your kids and Pickles too. Your care of him was wonderful and you seem to understand -- some pet owners don't -- that his lack of ideal cuddliness didn't make him a bad pet, it just reflected his tough life before you rescued him.
This is a time when your kids will start learning about death. The death of a pet very often is a child's first experience with death. When you tell the kids, you might want to emphasize that the vet did all they could; sometimes nothing can help; and yes, the idea of burying a picture, maybe along with a favorite toy of the cat's, could help. Or they could, in his memory, donate their pennies toward getting some cat food to take to a local animal shelter. That would let you show them that something positive can come out of pet's death. If you do that, be sure you are prepared for them to beg for another cat once they see the cats at the shelter, and decide in advance if you are prepared for that. I would definitely wait a bit before getting another cat, so that they understand you don't just "replace" a pet instantly and all goes on as before. But I would definitely have them do something positive and charitable for animals!