What's her level of understanding and following instructions generally? I'm guessing she's fairly high functioning from your description.
Sorry if it's an obvious question but is the idea of 'he might be put to sleep on the future' too abstract for her, especially in the face of how rewarding she finds picking him up?
Social story around how to look after dogs? With visuals if that helps her. There's a few online at varying levels
Perhaps a "don't touch" sign on his crate as a constant visual reminder so she can't 'forget'.
What normally motivates her? Can you try rewarding her with 5 mins of a favourite activity if she leaves him in the crate when you pop out. Maybe trial just going out for a short time (even if it's just 5 min walk to the end of the road) to start with and build up from there.
What does she like about picking him up?
Find something warm and fluffy to hold instead if it's a sensory thing? (hot water bottle with fleecy cover??)
Could it be redirected into 'looking after him' in other ways if that's what's rewarding? Getting his toys ready for when he wakes up or filling a Kong with treats etc. Could she look up tricks to teach him while he sleeps/isn't in the mood and then work on them at a set time each day? Does she usually play with him appropriately when he's awake?
In the short term, a padlock on crate so she can't get him out would help temporarily while you work on her behaviour. (It could be a combination lock if multiple family members need access) and then constant vigilance when he's out the crate
Sorry complete scatter gun of ideas there! Hopefully some might be useful
It sounds like hard work! 

depending on your preference