From my own personal experience, I feel your children will gain a lot from being w/Brewster in his final days or weeks. And Brewster will also gain from the love of those around him. He might not have lasted this long if it weren't for the love and care of those around him.
I remember losing a wonderful Fox Terrier when I was around 10 or 11. He died of hepititis. I still remember my mother holding me as I cried because of his passing. After a short time, they allowed me to pick another dog for a pet. I remember looking at 4 or 5 different puppies about 6 months old, all Fox Terriers and tho' I had recently lost Prince, my 1st Fox Terrier, I picked a puppy who was ill because I felt he needed some love and understanding even tho' I knew he would not be w/us soon. I didn't want him to pass away alone and w/out family. Surprisingly my folks allowed me to take him knowing they were paying for a sick animal. Tho' he wasn't w/us for long, I've never regretted choosing him.
I've discovered that thru' the years, all the dogs I've had who passed away are ALWAYS w/me and each one was special in their own special way. My 2nd dog's name was Silks.
Since then, in my adult years, I've lost 2 more and it doesn't matter if you're young or old, it's never easy to let go.
But I wouldn't miss having had the joy of being w/them and I'm sure that's what your children will remember thru' the years ahead as well.
I also noticed in your child's picture, that there are smiles on both faces. To me, this indicates that both child and dog are enjoying their time together however short it may be.