It's the only "fair" way to allow for everybody to be able to get these, I'm afraid.
They did the same exact thing with the Crash Test DummyHead (Cyborg Mk 2) on CO and Traffic Cone Hat as with these items. I earned those, but only the assimilated Tribble for STO. Kinda afraid to claim it since somebody said it was 1 per account and not 1 per character. Which is it guys?
They basically want everybody to be able to have access to this in one way or another, even though it spits on the testers who earned them and spent time on it.
It's one per character, not per account. You should be able to claim it on all your characters.