It's not quite the same but with our diabetic Pug my husband waited until I got home from a business trip. I got the sad task of taking our Pug the very next morning to be put down. The Pug had had enough and you could just tell by looking in his eyes. Technically our Pug started as my Pug. However, it really came down to my husband being way too soft hearted for the tough task. I know that about him so I wasn't surprised the job fell to me.
Oh and just to add. My MIL stored their Poodle in her freezer and I think no one ate out of her fridge for ages. She was totally unfazed, though. She said he was well wrapped and nestled among the tamales. No harm, no foul in her mind. You should see the suprised look on her face when people bring the story up and give her a hard time for her decision. To her he was a beloved family member, deserving of loving treatment even in death.
You know your mother better than any of us and maybe she just needs help with this difficult, sad task. If you feel confident she'll be okay with your taking charge of this task and by all accounts she seems to have given her tacit agreement with all the of paperwork she's provided, then take the dog in to be euthanized. Our vet clinic stored our Pug in their freezer until the following weekend. We picked him up the next Saturday to bury at the family ranch. The whole rattling Pugsicle on the drive to the ranch is another story entirely. Best wishes to you and your family. Good luck also.