The relay's orbital inclination can be changed so the plane of the orbit is perpendicular to the direction of the target star, so there's always line of sight. I think coherent light would be distinguishable from background sunlight.
I vaguely recall some telescopes simply block out sunlight by having a black disk in the center of the aperture, and orienting the telescope so the sun is behind the disk, but I can't find the search term.
That isn’t guaranteed to work if the relay is used as a crossroads on multiple routes.
And I think you mean a coronagraph (https://en.wikipedia.org/wiki/Coronagraph: “a telescopic attachment designed to block out the direct light from a star so that nearby objects – which otherwise would be hidden in the star's bright glare – can be resolved.“).
I’m not sure using these to block out the sun is common with very sensitive instruments. Some parts of Hubble can’t look at the sun as doing so would damage them (https://en.wikipedia.org/wiki/Fine_guidance_sensor#Hubble_Sp...: “Due to the sensitivity of the FGS they can not be used whilst the HST is pointed within 50 degrees of the Sun.”)
Also, of course, blocking out the sun’s light won’t make you see the stars behind the sun.
I vaguely recall some telescopes simply block out sunlight by having a black disk in the center of the aperture, and orienting the telescope so the sun is behind the disk, but I can't find the search term.