Earth already has one... or did. I'm not sure of the current state of what's probably classified anyway -- maybe LarryJ can fill in the blanks.

What I'm getting at is Cold War radar. NORAD, and before that the DEW Line. A bit more than 20 years ago, in a NASA publication on the state or SETI art at the time, I read that Earth was thousands of times brighter than the Sun at radar wavelengths. That, and periodic Doppler shift due to orbital motion, might be enough of a clue... especially to another civ' that had gone through a similar period of "duelling hegemonies".

Your laser idea, of course, also benefits from orbital Doppler to indicate it's on a planet, and isn't some weird phenomenon from the star. It suffers from not being omnidirectional...

Which brings me to the question: is the method intended to be "Hey, we're here!" to anybody in any direction - the equivalent of "CQ" to the stars? Or, like an observing schedule in reverse, does it illuminate different narrower parts of the sky, chosen (for instance) for the presence of nearby G stars?

No need to tell you, of course, but there's a three-way trade-off in signal strength (watts per area), vs the energy input and the technology to make it happen, vs the exepected outcome. Narrow-band signalling, even on a iron-clad regular schedule that re-illuminates the same patch of sky at the same precise interval runs a larger chance of missing detection if the listeners are (like we do) also looking at different patches at different times.

Back to your laser for a minute - can a laser be "chirped"? That would certainly add another indication of artificiality, if it's possible.