As the system you are asking looks to be a custom job, and you have the mandate to use some unknown "large button phone with 3 programmable buttons" I'd budget a hundred thousand to get it working. The developers I work with don't do pro bono which you didn't reveal your budget. This sounds like the button phone may need to go under the knife so to speak to be modified to work with this proposed system. This sounds like an expensive undertaking to me.

My dad when he had such a care schedule didn't need to do more than be at home for the care givers to arrive and do their work. Maybe such has changed over time but I'm a bit taken aback that such a system would be required today. There are a lot of aging folk and care systems out there and not one seemed to offer what you are asking for.

Good luck in your search.
Bob