As you mentioned, “the same snap works fine for crosswalks, set to entities.” which means you can able to fetch data for entity and that is the current behavior supported by snap.
As far I remember, Reltio supports fetching of entity /entities by crosswalk, not for relations.
Refer to https://help.reltio.com/index.html#relationsapi/dataapirelations.html for APIs for Relations which seems there is no API support for “Get Relations by Crosswalk”
While if you look at Entities section https://help.reltio.com/index.html#entitiesapi/entities.html , there is an API for same " Get Several Entities by List of Crosswalks" :
https://help.reltio.com/index.html#entitiesapi/getentitybycrosswalklist.html
You need to have Reltio Read (Object) snap for getting the relation where you do not need the crosswalk to search for.
