Is it possible to grind down river rock
By coincidence, I saw a video just the other day of an artist who takes river rocks and grinds decorations into them with a Dremel tool and a diamond bit.  So yes, they can be ground down.  

An angle grinder with a diamond grinding cup would do it, but like you said, it may dislodge stones.  A safer (but slower) approach is to use a Dremel tool with a diamond bit (like a drum or disk).  Feel around the surface for pointy areas and just smooth those spots.  The stones look polished (unless that's just a shiny sealant).  You can repolish the ground surfaces by using successively finer polishing disks (sets are sold for touching up granite counter tops; you can find them at large hardware stores or places like Amazon).

BTW, the grinding will produce a lot of stone dust.  Seal the drain so you don't clog it up.  Wear goggles and a dust mask.  The dust will be extremely fine, so clean it up with either damp rags or a vacuum with very fine filtration, like a HEPA vac.  A vacuum with a regular collection bag is likely to blow the dust out the exhaust.