You can clean a kriesel nursery tank the same as any other nursery tank. The fecal pellets produced by the newborns are heavier than water and sink to the bottom of the kriesel nursery where they could be siphoned from the bottom while making water changes using a length of airline tubing, just as you are accustomed to doing with an ordinary rectangular nursery setup.
The circular flow or "vortex" in a kriesel or pseudokreisel nursery tank is necessarily very gentle, not nearly strong enough to keep the negatively buoyant fecal pellets suspended in the water column. It keeps the live foods for the newborn seahorses (e.g., rotifers, larval copepods, Artemia nauplii) suspended in the water column and gently nudges them towards the center of rotation, because of their negligible weight and limited swimming ability, and the pelagic seahorse fry (which are neutrally buoyant) also tend to congregate at the center of this gentle rotation, which is where there food is concentrated. Both the live foods and seahorse fry are positively phototactic, and move towards the light, which keeps them away from the bottom, and the gentle circulation pattern concentrates them towards the middle of the kriesel nursery rather than at the surface.
But the heavy fecal pellets sink readily and quickly collect at the bottom of the tank as usual.
Best of luck keeping your nursery tanks clean and sanitary, Dave!