Didn't check carefully but this should be ok for ${}^\perp(\mathcal{W}^\perp)$ because $\mathcal{W}^\perp$ is closed under injective envelopes and everything is determined by the injectives it contains.

Any basic tilting object would have to be a pair of consecutive transjectives (being the only rigids, consecutivity to avoid Ext's) and so have to give n-Kronecker again. I am not sure if there's an argument that deduces it by first proving any derived equivalent algebra would have to be acyclic, but that would be amsuing.

Are you sure those are the only hereditary examples? Unless I'm misunderstanding something this is also true for n-Kronecker.