The laser was supposed to take care of SDI sats and other space-borne elements, indeed.

We may never actually know, but I have my doubts anything like that was ever fielded. You're talking about beam-expansion over thousands of km for a sat-to-sat weapon instead of the tens of km for the eventually de-funded ABL/YAL-1 (some two decades after the suggested Soviet space-based system.) You avoid the atmosphere in space but you can't avoid diffraction. Teller's original concepts for SDI space-based lasers proposed literal nuclear bombs to provide the necessary energy density at range; neither those, nor the other host of exotic 80s-era directed energy weapons concepts seem to have made it into operation.

i will have a hell of a time remembering where it was, but a few years ago i read an interview or essay that said this payload was actually a decoy, that the soviets were responding to the SDI program but could not get a workable laser with enough power in such a package, and that it was just there to look like a laser -- including by slowly leaking carbon dioxide, which would be the signature sought by american ground observers