How can a Muslim ensure that everyone is given their due right, and must he always satisfy all parties, or does this fall under blameworthy flattery?

Everyone is given their due right according to the Sharia, not based on the satisfaction of the parties. People's satisfaction is an unattainable goal, but if it is possible to achieve the satisfaction of the parties without violating the Sharia or harming anyone, then it is a good and desirable thing. This is evidenced by what Hani' did with his people, as he would reconcile between them, and both parties would be satisfied, which the Prophet, peace and blessings be upon him, approved of.

Sheikh Saleh Al-Fawzan clarified that this reconciliation, which ends disputes and satisfies both parties, is a good deed.