Rights exist independently of society. Take a man and remove him form society, and he has the right to do anything and everything he can possibly imagine.
What is a social invention is the ability to implement consequences for exercising these rights which exists.
In any society, a person has the right to do whatever I wish, regardless of what that is. What society has invented is not the rights, it is the consequences that can exist for exercising these rights. But so long as it remains possible to exercise those rights, they exist regardless of those consequences.
The only way to alienate a right is to remove the ability to exercise that right. It is not simply creating consequences for exercising those rights. Thus, if a person has the ability to exercise a right, it is inalienable. Society cannot remove the person from their ability to exercise a right.
What has been decided by convention is that certain rights exist that the social authority should never create consequences for. These rights are those that should be freely exercised, without the threat of consequences.
Society can, however, decide which rights they want to prevent it's people from freely exercising. An example of this is taking intoxicants. I have the right to do this, and no law exists that removes my ability to do this.
What society does is prevent me from exercising this without risking consequences. It cannot alienate me from this right, though. It could only do this by removing my ability to engage in that right, but since that is impossible to do while I exist, the right remains inalienable.
So rights are inherent, and they are the natural default state. What is a social construct is any consequences for their exercise. Rights exist independently of society, while consequences for engaging in a right exist only because of society.
A social construct cannot exist without society, and therefore, rights cannot possibly be a social construct.
Government is a social construct, that exists to place limitation on the free exercise of rights. What a bill of rights does is place limitations on the government's authority to limit the free exercise of certain rights.
It does not place a limitation on the rights -which exist independently of convention- it simply places a limitation on the potential consequences for exercising one's extant rights.
One does not need to believe in a deity to realize that logically, the existence of all rights is the default state of man.
Remove society, and there are no limitations to the rights one can freely engage in. Obviously, that means rights exist independently of society.