OSM policy for names is to use the name that is used locally (but with a preference for signed names). Both names are signed in this case but the more locally used name is likely to be the English language name.

Currently, there are under 1,000 people that speak the Halq’eméylem language whereas there are 100,000's of people that speak English in Vancouver.

In addition, other government services, such as Canada Post and the City's own webmapping are using the English language name.

If someone feels strongly about this, I recommend first reading the Gulf of America controversy on the OSM Community Forum. This provides a good insight into why official names aren't always the default name in OSM.

If you still feel really strongly about this then I recommend moving to a dual naming structure, such as "Musquemview Street / šxʷməθkʷəy̓əmasəm". 