I have no authority on the subject, but I went through Quartet 1 directly after Genki II and before even starting Bunpro and was just fine. No textbook is going to have every single grammar point in it. For example, I’m doing the N3 新完全マスター book right now and it has grammar points Bunpro considers N2 in it.
I think you should evaluate your current situation and goals and choose an option that most closely matches. If you aren’t planning on taking the N3 until next December, why not slow down and finish out the N4 Bunpro vocab and Grammar tracts?
On the other hand, I am going through and completing the N3 vocab deck before starting Quartet II because 1) I am not taking the N2 until next December and am not in a rush and 2) Because the vocab used in Quartet 2 readings is much, much more complex than that in Quartet 1.
I worked through Quartet 1 while also learning a few Bunpro N4 grammar points a day and that worked fine for me. As always though, one size doesn’t fit all. Happy studying!
TLDR: Pick what works for you, but all things considered you won’t have any trouble with Quartet 1 content if you have finished Genki II.