Sentient rights, sapient rights, legal resident rights, citizen rights...
There are basically two things you need to consider here.
First, there is no single correct option. This is because these "fundamental rights" (another option) are not actually "fundamental" rights, they are legal constructs granted by a political process. The correct word to use is the one that properly describes that political process and the reasoning behind it. So to pick the right word you need to make up the history behind how people got those right and why and then (sorry about this) briefly try to think like a politician selling the idea.
Sometimes, surprisingly often really, the terms that stick were originally coined by people opposing the idea as derogatory terms. This is even more likely if the lofty ideas have a history of implementation problems. So you might want to think about it from that direction as well.
Second, The legal rights are not a single monolithic object. For example, while there is almost infinite number of possible terms you could choose, the reason I picked the four in the beginning (sentient rights, sapient rights, legal resident rights, citizen rights) is that they represent a continuous scale of scope. There are sentients that are also sapient and have sapient rights in addition to their sentient rights, but all sapient beings are also sentient and automatically have all the sentient rights in addition to their sapient rights. And so on. So you would want multiple different grades of rights with separate names and legal and ideological basis.
Historically and realistically in addition to such logical and neatly scoped rights you would also have a patchwork of more speficic rights and privileges granted in a specific historical context without any overlying design and logic. Say a specific group of refugees or immigrants or a religious or ethnic minority might get a special deal in response to a political crisis and that deal will then be on the books for decades or even centuries. You might want to add one or two such special cases to your system to make it look more realistic. Sadly, messy -> realism.