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This will be possible with the Project tool with ArcGIS 10.4 (already available as pre-release). An additional parameter 'Vertical' allows you to specify an input and output vertical coordinate system, in addition to the horizontal coordinate system: This new parameter is mentioned in this What's new in 10.4 document (without much detail).


This should answer your question:


Is the raster format an Esri GRID? If so, the original LCC definition may have used NAD83 (CSRS) which isn't supported in the PRJ.ADF file format used by Esri grids. The definition doesn't match any of the existing Canadian LCC definitions, but it's close. Do try to change the geographic coordinate reference system only to NAD 1983 (North American 1983) ...


It looks to me like the Lambert projection file has an error. GRS_1980 is a spheroid, not a datum. The NAD83 datum is based on the GRS_1980 spheroid. I think it is possible that your Lambert projection is actually using the NAD83 datum. Your datum transformation throws an error because you are not trying to transform between WGS_1984 and NAD83. If it were ...


The offset that you have between your bathymetry and your elevation model relates to the difference between the Mean Seal Level (MSL) and the Lowest Astronomical Tide (LAT). From the well written text found here we can read that: The MSL surface is in a state of gravitational equilibrium. It can be regarded as extending under the continents and is a ...


A great tool that is often used for vertical datum transformations is called V-datum, developed by NOAA. This is the tool that I have seen cited most often for similar tasks with a range of different datums including tidal. The V-datum docs has some information, as well as this gis.stackexchange answer on EGM96. I'm not familiar with lowest astronomical ...

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