Regarding your immediate issue, the workaround I see would be that you use PROJ to create a regular grid that contains the vertical correction, use gdalwarp to transform it to the exact extent and resolution of the result of your 2D-only gdalwarp, and finally use gdal_calc to sum the two.

Well, if you actually create a vertical correction only grid, then you could just use  -s_srs "+proj=longlat +ellps=... +geoidgrids=your_tif_or_gtx"  (or maybe as output, or fix the sign of the correction in the grid), since the workaround I give above is what gdalwarp would do. But of course none of this will solve the time-dependent aspect you need.

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http://www.spatialys.com
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