The increasing statistical significance of the DAMA/LIBRA annual modulation signal is a cause for tension in the field of dark matter direct detection. The COSINUS experiment aims at a model-independent cross-check of the DAMA/LIBRA signal claim, using NaI crystals operated as cryogenic scintillating calorimeters at millikelvin temperatures. Such a setup enables measurement of phonon and scintillation light signals via Transition Edge Sensors (TESs) and allows particle discrimination on an event-by-event basis. The non-standard properties of NaI cause an obstacle when attaching a TES directly onto the surface of the crystal. This can be overcome with the "remoTES" design, where the TES is attached to an external wafer crystal. We present the results from a first successful operation of NaI and other crystals as cryogenic calorimeters with the remoTES design.