As others have said it is Julie & her husband's decision and, that being said, whatever decision they make will be the right one for them & for Maverick.
I'm with Dreeza as far as the bashing of the mother and the calling of her children spoiled. Being a mother & grandmother I can thoroughly understand wanting to give your children what you can that will bring them happiness and to try to keep sadness away from them if it is within your power. I think that is what this mother was trying to do. Missing posters convey an owner is accepting responsibility for the missing pet - it doesn't necessarily mean love and a true desire to have the pet returned - it does show the desire to assure the well-being of the pet.
Many years ago I had a litter of kittens. A young mother and her son came to look at the kittens as the cat the mother had had for years was lost due to old age. Her young son had grown up with the cat. When he walked into my house he immediately starting smiling and said "Mommy, here's our cat!". The mother cat looked identical to the cat they had recently lost. He had no interest whatsoever in any of the kittens - I brought each & every one over to him and explained they were the ones looking for a new home, not the mommy cat. He started crying (he was only about 5 yrs old). I started crying right along with him as did his mother while my cat was snuggled up just as cozy as could be in his lap & purring her utter contentment. I didn't have the heart to deny him and I knew Blackie would be totally spoiled and well loved. That day they left with Blackie and my phone number. For the next 10 yrs I'd get 1-2 calls a year from Timmy to tell me thanks and give me an update on how Blackie was doing. He also called me when Blackie passed away and thanked me for letting him have her to love for all those years.
Was that mother being unreasonable and was Timmy spoiled? I don't think so! Was I an owner who really didn't want her cat? No!
Would I do it again? I don't know - it would totally depend on the situation I would imagine. I do pray that I never find myself in that situation again as it is a hard one to be in.
OK, I'm off my soapbox and I hope I didn't offend anyone - that was not my intention.