pretty sure you could do a private crucible match and it will drop, or simply play some crucible and it will eventually drop.

I just got the catalyst for it today after running The Coil on Expert

I’ve been getting the strikes ones from fireteam ops expert and higher

Prior to edge of Fate competitive and nightfalls. I recall hovering over the comp node to see it saying exotic catalyst were rare. Also in the catalyst section of collections it used to tell you where each catalyst could be obtained. Nowadays in edge of Fate I'm not certain if these methods still work, as I already have every catalyst done and finished. If I see something new I'll be sure to return with better info. Hope the methods stated for prior to edge of Fate work still, keep in mind the catalyst that do drop are random until you've collected all of them. It'll be easier to get new catalyst if you already have em all when a new one becomes available outside of an exotic quest for it.