I received the Jade Rabbit catalyst quite a while ago. Never bothered to complete it. Momentum Control this week has been the perfect time to work on it, and got it done today. I'm on PC.
Now, I reckon I know where the problem lies. Back a while, you used to be able to go into the Destiny app on iPhone and select a kill counter for weapons that weren't masterworked, but that bug was patched out a long time ago. I did that with my Jade Rabbit. Here, (hopefully I linked the image right) you can see that I am unable to choose the PvE or PvP kill tracker, but it still has a kill counter. Don't know if this is important or not, but I applied the completed catalyst during the middle of a game of Momentum Control.
I had also applied a kill counter to my Ace of Spades but that one was fine when I got (and completed) the catalyst.
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Jade rabbit doesn’t get the option to change between the separate counters.