But you are, in fact, begging the question. What you say is that you cannot be a positivist and account for rights. But this rests on your definition of rights. Put very simply, your argument is "Rights emanate from a divine source, therefore you can only account for rights as emanating from a divine source."
A positivist definition of rights, simply put, goes like "Rights are affirmatively assented to by law." Boom, that's their accounting of rights. If you want to dispute it, that's all well and good, but you're equivocating on two very different issues, the argument in favor of the natural rights theory on the one hand and the argument that positivists simply cannot account for rights by definition.
Here we are only concerned with your latter point, which hinges on your very attenuated and obscure definition of rights, which you can only arrive at by way of your larger point in favor of natural rights theory. We're not concerned with your larger point, we're only talking about what positivists say and what naturalists say. They both talk about the same rights and the same equality, they just have different ways of getting there.
If you think that the positivist account is not convincing, it doesn't really matter to this discussion, I'm actually a proponent of natural rights theory myself. But since you are also arguing that the positivist account of rights doesn't exist at all because it conflicts with your definition of "right" then you are begging the question.