What is the ruling on interrupting the Eid takbirat to say hadiths such as: "Allah does not forgive two who are at loggerheads," and "He who is deprived of gentleness is deprived of all good," then resuming the takbir?

It is recommended to openly pronounce the Takbir during the two Eids in mosques, homes, streets, and markets. People can be reminded of some meanings while pronouncing the Takbir when needed, as long as it does not become a regular Sunnah. The original place for these reminders is the Eid sermon.