YNK You seem to be reading a different thread to everyone else
Of course the OP would be the one with ultimate responsibility for the dog. It's meant to be therapeutic so I don't believe the OP would set her up to fail.
I'm sure nobody would intentionally set their child up to fail, but getting a dog is not the only answer ... it could help or it could not, there are no guarantees. One of my friends got a dog as she was advised it might help her depression ... only when she had a deep bout of depression that meant that she barely got out of bed, having to feed, walk, pick up after the dog as well was just an added pressure that she couldn't cope with and she ended up being more heartbroken when she had to admit she couldn't cope with the dog and had to have him rehomed for his wellbeing and her mental health.
I'm sure the OP understands that having a dog would provide an opportunity for the parents to demonstrate and teach their DD that there are rewards for loving unconditionally even though it involves making some personal sacrifices.
Or it will provide an opportunity to show that people should ignore the obvious symptoms and sacrifice their own health because of someone else's whims and wants.
There's making sacrifices and then there's affecting someone's health - the OP's DH has both expressed his wish not to have responsibility for a dog and has a good reason in having allergies, and yet both of those are being ignored because it "might" help DDs mental health. Might affect MH does not trump the fact that it will affect the DH's physical health.
It looks to me like its her DP who is prepared to sabotage any therapeutic benefit for the DD by his selfish need for control
It is not a selfish need for control to not want to have to suffer allergies every day for the next fifteen years! Not to mention the DH has actually agreed to have a dog - on a fixed trial basis - to see how it goes. I think that is a very selfless and sensible idea.
The DD is not a baby, it could be clearly explained that this is a trial situation, with a dog going to be an Assistance Dog, and depending on how that goes - with DHs allergies, with how it helps DD, and whether or not the novelty wears off - they can then consider whether it's a viable long term solution.