There are plenty of passive ways to gather the same data, that don't broadcast our existence into a galaxy we know nothing about. Let's use those first, and after we learn, decide if we want to talk back.
We've got millions of years to figure out what to do. There's no rush. It would be amazing to finally know we're not alone. That's a great first step.
However one interesting idea WRT K series, is as a dwarf, I bet a "nearby" sufficiently bored, yet paranoid, civilization could mess with a K series by periodically blasting it remotely with a truly huge laser/maser and mirror system. I'm not talking about bored ham radio operator level of commitment but a suitably high tech advanced civilization could possibly pull this off. "Well, we're building a solar system wide asteroid defense system anyway, so letting the scientists do research 1% of the time when nothing else is going on, won't hurt too much"
So possibly the G series civilizations are rather psychologically optimistic, or pessimistic about physics anyway, and K series stars are reflected/enhanced light from a troll civilization located within 500 light years or something. Perhaps the K series stars are reflectors for a nearby G series civilization.
In that case maybe the ratio of G/K reflects some kind of galactic intelligence level of average paranoia.
Or for that matter vice versa. Perhaps we're the odd men out and most galactic civilizations are K series and find it hilarious to "frame" innocent G series stars.
I suppose extremely detailed planet searches will be performed on these interesting stars, regardless of natural or SETI cause the results are likely worthwhile.
Very possibly every other intelligent species in the universe has figured this out already and that explains the Fermi paradox. Species that were too dumb to figure it out have just become extinct, so we don't hear them either.