I feel left out.
I accidentally accessed Klingon on my android-type phone, and got to do one lesson. I thought, "All right!" But after the lesson, the menu continued to lack Klingon. So why can't I do Klingon on my phone?
Other than quite a few bugs involving capital letters compared to the web-based version (which I usually use), Klingon works fine on my Android ap.
You may want to try rebooting your phone and trying again.
If that doesn't work, perhaps make a bug report here: https://support.duolingo.com/hc/en-us/requests/new
This was posted today on the Facebook KLI page:
"if any of you, like me, has a phone with an old android version where the Duolingo app does not allow the Klingon course, I found a workaround. there's something called, I guess, PWA (progressive web apps), which lets you add a "shortcut" of a web site that works almost like a real app. it's available in Firefox, next to the address box, as the silhouette of a house with a plus sign. I guess it is possible in other browsers."