Is it permissible to perform Dhikr—such as saying "Lā ḥawla wa lā quwwata illā billāh" (There is no might or power except with Allah), or "Astaghfirullah al-ʿAẓīm" (I seek forgiveness from Allah, the Almighty), or "Allahu Akbar" (Allah is the Greatest)—with the intention of admonishing or hinting to someone to stop a certain action? And does this contradict the primary purpose of Dhikr?

There is no harm in remembering Allah (dhikr (definition) Allah) when expressing wonder or finding something extraordinary, for it trains the tongue to remember Allah. Imam al-Bukhari titled a chapter in his Sahih (definition): "Chapter on saying 'Allahu Akbar' (God is Greatest) and 'SubhanAllah' (Glory be to God) when expressing wonder," and he cited three hadiths under it. Using these expressions when marveling at something or finding it extraordinary is good. There is also no harm in using these invocations outside their primary meaning, for other purposes, and this was common practice among the Salaf (pious predecessors).