Sanded grout is what you need
Sanded grout is used when grout lines exceed 1/8”. These pebble mosaics by their very nature have grout spaces that exceed that dimension a few times over. To maintain the desirable bumps of the pebbles, the grout needs to be applied at a depth that allows the stones to assert their personality.
When I grout for these parameters I like to work in smaller areas so I can stay on top of the grout clean up, just like when working with thin set grout has a limited “pot life” and when it starts to become stiff the ease of workability does down. Most grout cleanups are done pretty dry, with a grout sponge that is barely damp. When working with pebbles a wetter sponge will allow you remove a bit more of the grout to maintain the profile of the stones' bumps.
Simply running a grout float over the stones and doing a quick wipe with the sponge results in a pretty flat and smooth surface…but if you wanted flat your should not have spent all of that extra money on the pebbles.