How the seafloor shapes the water surface
Bathymetry is the underwater equivalent of topography: the depth and shape of the seafloor. It matters to the surface because waves slow down and bend when they reach shallow water. In the North Sea, shifting sandbanks in an ebb-tidal delta cause non-linear wave propagation, which is why a wave-height prediction model needed bathymetric data as an input to be accurate [2]. In that setting, the same offshore wave field can produce different significant wave heights depending on what the seafloor looks like beneath it.
The effect is not limited to waves. A high-resolution bathymetry and topography dataset for the Amazon estuary was built specifically because characterizing estuarine hydrodynamics depends on knowing both the underwater depth and the surrounding land surface [5]. That is a statement about what is needed to model water movement, not a measurement that bathymetry alone sets the average sea level.
Why satellites can read the seafloor from the sea surface
The link also runs in reverse. Satellite altimetry measures sea-surface height, and those height measurements are used to recover global marine gravity fields and seafloor topography models [4]. In other words, the ocean surface carries a faint imprint of the gravity field shaped by underwater mountains and trenches, and that imprint is what makes bathymetry mapping from space possible.
The same logic appears in rivers. The Surface Water and Ocean Topography mission measures water surfaces, and researchers used those height observations to estimate river bathymetry, which cannot be measured remotely at large scale [3]. On a small UK river, the resulting bathymetry estimates were accurate enough that flood models built from them simulated flood extents and depths about as well as models based on local river surveys [3]. That is evidence that water-surface height contains usable information about the bed beneath it.
What this means for the ocean's height
For the open ocean, the practical answer is that bathymetry affects local water height mainly through the movement of waves and tides rather than by raising or lowering the ocean as a whole. The clearest evidence supplied here is coastal and estuarine: wave propagation over changing sandbanks [2] and hydrodynamic modeling that requires both bathymetry and topography [5]. These are local and regional effects, not a global sea-level signal.
There is also a modeling caveat worth keeping in view. A seismic tomography study of eastern Indonesia accounted for topography, bathymetry, and the fluid ocean explicitly and examined how surface elevation affects synthetic seismic waveforms [1]. That work treats bathymetry as something that changes how signals travel through the Earth-ocean system, which is a different kind of influence from changing the ocean's standing height. The evidence supports a two-way link between seafloor shape and water-surface height, but it does not show that bathymetry sets the ocean's average height.
Sources used in this answer
<i>SASSIER22</i>: Full‐Waveform Tomography of the Eastern Indonesian Region That Includes Topography, Bathymetry, and the Fluid Ocean
A full-waveform tomography model of eastern Indonesia incorporated topography, bathymetry, and the fluid ocean, and examined path-dependent effects of surface elevation on synthetic seismic waveforms.
Spatial ocean wave height prediction with CNN mixed-data deep neural networks using random field simulated bathymetry
A deep neural network predicted nearshore significant wave height using simulated bathymetry, because changing sandbank morphology causes non-linear wave propagation that requires bathymetric input.
Flood inundation modelling using the Surface Water Ocean Topography Mission
Methods were developed to estimate river bathymetry from Surface Water and Ocean Topography mission height observations, and the resulting bathymetry was accurate enough for flood modeling comparable to survey-based models on a small UK river.
Research progress of ocean satellite altimetry and its recovery of global marine gravity field and seafloor topography model
A review of ocean satellite altimetry describes how sea-surface height measurements are used to recover global marine gravity fields and seafloor topography models.
Comprehensive bathymetry and intertidal topography of the Amazon estuary
A high-resolution bathymetry and intertidal topography dataset for the Amazon estuary was created because characterizing estuarine hydrodynamics depends on knowledge of both bathymetry and topography.