Is it permissible for a grandfather to use a power of attorney from his deceased son, after his the death of this son, to sell his son’s share in something to his other son?
To what extent is this considered to be a legal contract?

A power of attorney is an instrument authorizing another to act as one's agent in things which are possible to undertake on another's behalf during one's lifetime. It is legitimate, since Allah the Almighty says:
Appoint one arbiter from his family and one from hers.
[4:35]
Authorizing another to act on one's behalf is likewise permissible based on what was narrated that the Prophet delegated messengers to collect zakat. In addition to this, the Messenger of Allāh appointed ‘Amr Ibn ‘Umayyah al-Damri to act as his agent in his marriage to Umm Habibaba.1

It is a prerequisite that it be valid for the grantor to engage in transactions. A power of attorney becomes invalid when either the agent or the grantor ceases to be qualified to engage in transactions, through death or insanity. So it is not permissible for the agent to engage in what one was commissioned to do after the death of the grantor.

The ruling

Based on the above, it is not permissible for the grandfather to use the power of attorney to sell the share of his deceased son—the grantor—to his living brother. If this does occur, the sale is legally void.
And Allah the Mighty knows best.