I disagree with Caliban and most others, here. I think you need to evaluate the rules of the particular illusion and the particular form of sensing and se how they interact. Here, we have an illusion that is specified as very powerful - it impacts all creatures (that can perceive the illusion).
The more specific rule wins. The text says: "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".
Illusions, by their descriptions, can have more than just visual elements. As such, assuming that a creature with blindsight is not impacted by all illusions is not a safe assumption. The blindsight, whether it is based upon echolocation, sonar, or another means, can be fooled by an illusion if the illusion says that creatures with blindisght are impacted, and this magic item specifies all creatures, which would include those with blindsight, are impacted.
If a creature were specifically immune to illusions, the specific immunity to illusions would override the cloak as the cloak explicitly works via illusion, but unless they added language to blindsight in some place that specifically provides immunity to illusions, blindsight is not immune to all illusions.
I would also rule that an illusion that is specifically limited to only visual elements, such as minor illusion (visual) that is specifically limited to only a visual image with no sound, light, smell, or any other sensory effect, would be something that blindsight looks past. As long as you have normal vision in addition to blindsight, you would see right through it without the examination. Why? Blindsight specifies that you can see without relying upon sight and the only thing that is there for a minor illusion is a visual effect - meaning you can inherently see with blindsight that it is not there.
We can argue about what "appear" means in the cloak description, but appear, by dictionary definition, is not strictly a visual term by all possible interpretations. It can refer to being noticeable by any means, or to seem (based upon any criteria).