One of the problems you face in this question is getting around the fundamental presumption of competence that is so deeply ingrained in you that you don't even notice it. Despite what a lot of other people have said in the various comments, human justice has never had a problem determining that for the most part humans are responsible for what they do. "Marginalized groups" would be ones who were perhaps excessively punished for what they did, or punished for things they did not do at all, but there's nowhere in history that a judicial system has ever had the slightest problem condemning a human for murder or something. Casual assumptions about humanity have been made since we are the only species that can really be expected to act "responsibly" for any common definition of that term.
Likewise, there's no justice system of significance anywhere in history that has treated animals as full individuals, because they can't be expected to be responsible for anything. If nothing else, even the very greatest animals we know show very little capacity for understanding what may happen a day or two from now if they act in a certain way today, which is a very important element of holding people responsible for lawbreaking, since only the very most basic, most brutal elements of law involve consequences immediate, large, and guaranteed.
I would suggest that the proper way of evaluating such a beast would be seeing to it that rights are matched to responsibilities. As humans, we have the right to not be murdered by other humans; we have the corresponding responsibility not to murder others. We have the right to security in our properties, and we have the corresponding responsibility to not steal from others. And so on. The question would be, is this individual capable of discharging the responsibilities associated with the rights we are considering giving them? Someone who is literally incapable of not hurting other people is not put in jail, they are institutionalized.
If they are not at all capable of handling the responsibilities, than while we may be forced to take certain actions in regards to their crimes, as we might put down a dog that violently killed a child, we would not necessarily consider them culpable. Suppose there was a species that was mostly like humans, but if you punch them in the grondar, it can be scientifically demonstrated that their forebrains are turned off and they violently attack anything in the area. They lack all ability to not do this. It doesn't matter if they train, or meditate, or whatever. If they get punched in the grondar, we can't hold them morally responsible for what happens next. Then again, perhaps they are supposed to be wearing their grondar-protectors and they left it off today, in which case, yes, they were capable of discharging their responsibilities and they failed, and now they are responsible for what happened.
The other side holds as well, too; non-human sentients can have both rights and responsibilities that we don't. Humans are not necessarily the pinnacle. The real human world already has exemptions for "fighting words" and "crimes of passion" (i.e., catching your mate mating with someone else), for which we will reduce the perceived culpability of someone for a violent action because of the provocations that we are ourselves quite vulnerable to, and it becomes unreasonable to expect the entire population to be able to uphold the responsibility of staying calm under those provocations. But perhaps Vulcans still are expected to be that calm. They may have some corresponding rights that humans do not. Or in a post-Singularity science fiction story, unmodified humans may not be held responsible for actions taken after interaction with a Class 3 or greater artificial intelligence, if such AIs are well-known to be able to play unaugmented humans like fiddles if they choose to. Some thought may deal with areas in which humans may already be not held responsible.
With some creativity, you may even come up with situations in which it isn't clear that humans are "deficient" per se, they just aren't responsible. For instance, perhaps sentient dogs could be punished for "disloyalty to the pack". Humans are not generally punished legally for this, because our loyalty relationships are much different than pack relationships; not necessarily worse, not necessarily better, just different. The sentient dogs may have rights and responsibilities that are simply meaningless for humans. By contrast, as we punish humans who promise to protect confidential information and fail to do so, perhaps the sentient dogs can only be cleared for security work whole packs at a time, because there is no way to have a sentient dog loyal to an abstract organization above their pack. etc.
One of the casual assumptions we make because everyone is human is precisely that the question is binary; are you competent or not? Human/adult or not? This makes sense in the human world, and is probably even a good idea to prevent the creation of distinctions where they don't really exist (i.e., racism, etc.) In a multi-species society, the question may be more granular than that. The question you may be asking may not be "is this creature a legal person?" but "is this creature capable of not doing the thing we're calling a crime?"