Lyxen
Great Old One
For the first time I realised that Cloak of Displacement doesn't say if your REAL image becomes invisible...
It does not need to: "While you wear this cloak, it projects an illusion that makes you appear to be standing in a place near your actual location, causing any creature to have disadvantage on attack rolls against you."
As the name implies, it displaces your image, so it's not at the real location anymore, it's at the new one. It's not creating a new image like mirror image, it's using your image, just appearing to be standing a bit off.
I think the RAI of "displacement" is definitely that you are in one place but everyone sees you a bit far away, but someone might claim you see double.
I don't think someone could reasonably make that claim based on the sentence above.
Anyway for the question at hand, I would check if the opponent with blindsight also has normal sight. I think normally blindsight is given to creatures without eyes, in which case I think they will just know your true location and would not see the illusion at all. If they do also have normal vision (maybe someone who is getting blindsight from a spell or item) then it might be interesting to consider letting them see BOTH and be puzzled for a bit not knowing which one is right, but then there are spells like Blur and Mirror Image which reveal that probably the RAI around blindsight is to beat this sort of effects.
I think it really depends on the type of blindsight. 5e has a general blanket description for blindsight, but you can imagine different types, from echolocation to heat to just mystic feeling. Moreover, the illusion type is not precise either, nothing says that it displaces only the visual image. It makes you appear a bit off, it might be complete with sound, heat, etc. in which case it might fool some types of blindsense.
At this stage, it's really a DM's call, 5e is not precise enough.