I'm so sorry about your dog's illness.
Working with a trainer to raise and train a service dog rather than going through a service dog organization is a feasible option, but given what you've shared here I'd consider a few things before going ahead with this plan:
1 - are you already in contact with a trainer who has proven expertise in this area and is willing to take you and a pup on as a client at a rate you can afford? This is a very specialized training task, not something a typical dog trainer would have done before. I wouldn't get a pup until I had this lined up.
2 - since you'll need to take the dog everywhere with you, you might consider a smaller poodle than a standard. They are more welcome in public places and easier to travel with. Smaller dogs can be quite successful as service dogs for certain tasks, including medical alerting.
3 - I'd be iffy about going with a poodle that's bred to do agility specifically. Poodles are pretty high energy, rambunctious dogs to start with even before you add that sort of specialization. I have a two year old poodle from hunting lines and he's a whirling dervish. For a service dog, I'd be looking more for a poodle from a sire and dam that are excelling at, say, therapy work and/or obedience, if I couldn't find poodles being bred specifically for service.
4 - have you considered what you'll do if as it grow up, the puppy washes out as a service dog for whatever reason? Even with the best genetics, planning and training, many dogs just don't have the aptitude or the temperament for it.
Working with a trainer to raise and train a service dog rather than going through a service dog organization is a feasible option, but given what you've shared here I'd consider a few things before going ahead with this plan:
1 - are you already in contact with a trainer who has proven expertise in this area and is willing to take you and a pup on as a client at a rate you can afford? This is a very specialized training task, not something a typical dog trainer would have done before. I wouldn't get a pup until I had this lined up.
2 - since you'll need to take the dog everywhere with you, you might consider a smaller poodle than a standard. They are more welcome in public places and easier to travel with. Smaller dogs can be quite successful as service dogs for certain tasks, including medical alerting.
3 - I'd be iffy about going with a poodle that's bred to do agility specifically. Poodles are pretty high energy, rambunctious dogs to start with even before you add that sort of specialization. I have a two year old poodle from hunting lines and he's a whirling dervish. For a service dog, I'd be looking more for a poodle from a sire and dam that are excelling at, say, therapy work and/or obedience, if I couldn't find poodles being bred specifically for service.
4 - have you considered what you'll do if as it grow up, the puppy washes out as a service dog for whatever reason? Even with the best genetics, planning and training, many dogs just don't have the aptitude or the temperament for it.