Hi there, I think it would be more consistent with how other honorary street names are tagged to do this:
name=Alaskan Way
name:en=Alaskan Way
alt_name=Dzidzilalich
alt_name:lut=Dzidzilalich

Does this change seem okay? I think it's worthwhile because Dzidzilalich is not a translation of Alaskan or Elliott Ways but rather has a separate meaning (unlike the streets in the CID which are have signposted translations).