How do we perceive weight?
This would most likely come under the category of Proprioception - the sense of how our bodies are arranged in the world, and how much effort each part of our body is expending.

From Wikipedia, 'In humans, it is provided by proprioceptors in skeletal striated muscles (muscle spindles) and tendons (Golgi tendon organ) and the fibrous membrane in joint capsules. '.

So our brain receives signals from muscles, tendons and joints, and integrates these into an internal sense of how we are arranged and what our bodies are busy doing.