Can't government experiment on price in the case of a monopoly?
If I understand your question correctly, you are proposing that the government sets prices and observes the behavior of market actors to elicit the demand and supply functions.

I see two possible concerns: 
Unless the monopoly is obligated to serve all customers, the market may not be in equilibrium, full demand may not be observed.
Similarly the government would have to buy any unsold goods, otherwise
a too large $p$ could result supply being larger then demand, resulting in a loss for the monopoly.

The other concern is that the monopoly's behavior significantly affects the government's observations. It could potentially benefit by behaving suboptimally now and earning more in the regulated future. One way it could do that is to make the government believe it has higher costs by supplying lower quantities given any price $p$.
-While I was writing this answer, @Dan has posted the same idea in a comment.