Service dog puppy or organization

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.

@parus Just a guess but if OP has balance issues a larger dog may be a better match if they want them to perform some bracing work.

I know some medical alerts are extremely tricky to deliberately train if the dog doesn't alert naturally. I don't know enough about it to say where blood pressure falls on that scale, but I'd definitely be looking for a service dog trainer or support group to reach out to and see what the process will be like. I would definitely want to know for myself what the process will look like if your new SD prospect doesn't develop a natural alert.

I'd absolutely prioritize a breeder who's produced successful service dogs, or at least dogs who do work or activities that require a very stable, calm temperament and easy acceptance of strange things or busy environments, like therapy dogs who visit hospitals and care homes, as Parus said. Sports like agility tend to call for a quite different temperament and an intensity that might not translate well to being calm and focused working in public spaces and ignoring distractions.

A lot of people in the US do owner-train their service dogs, with or without a professional trainer's help, due to how depressingly difficult it can be to get a program dog, especially for people with unusual or very specific needs. It's definitely legal, but challenging and a long process. I urge you to look for online communities specifically for service dog teams and trainers - we'll of course try to help any way we can here, but we're a diverse group and most of the currently active members can't provide the same kind of advice and experience you can get from other US SD teams or people training these dogs professionally. Facebook is the best place for these kinds of groups right now, but there's likely more specialized forums out there as well.