I just saw your addendum. What a wonderful answer! Thanks a lot. This completely explicit construction goes above and beyond what I expected.

Thanks very much! This confirmed a few suspicions of mine. When you have the time I'd love to see all the other examples you mentioned, but this is already pretty good :)

@RobertBryant thanks a lot! However, I must apologize again for not having included the hypothesis of dimension $\geq 4$ in my original post when I mentioned "part of some work that I've been doing with some other people...". But I am indeed very interested in knowing the explicit example of this $3$-parameter family you mentioned, I would very much appreciate it if you shared it.

@RobertBryant sorry, I indeed forgot to add it as a hypothesis. I'll fix it, thanks for pointing it out.

Thanks very much for the answer, but I would prefer examples where $\lambda$ is not constant. I'm sorry I didn't specify that, I'll edit my question.

@DeaneYang I guess it was a mistake on my part to call it a book. The link is correct. The page I referred to is page $165$ of the PDF and page $329$ of the article (where the first estimates I cited were proved, which the authors then claim in the corollary $5.2.7$ (located at the end of page $168/332$) implies the estimate I'm having trouble with

@WillieWong thanks a lot! That settles it then. The book is indeed working with Ricci-flow in 3 dimensions.