If your goal is to be able to change a measure name in the SSAS model and have that name change update all the reports built from that model, then I think you're out of luck since the measure name is its identifier (as you've noted).
On the other hand, if you're working with a specific report connected to an SSAS model and you want to display a different name for the measure, then you have more options.
The simplest is just to relabel the measure in the Values area of the visual to change how it appears in that visual. This renaming is localized to only that visual.
Create an alias measure as you've suggested in the report you're working on. The alias measure is localized to the report it's defined in. It can be reused across multiple visuals and pages of the report.
Create an alias measure in the SSAS model. This alias measure is then available to any report that connects to the SSAS model.
I can't think of any reason why creating alias measures would cause performance issues. It could potentially make things a bit more confusing having multiple measures for the same calculation but you can probably mitigate that by e.g. using display folders.
Thank you Alexis for your helpful answer.