It seems .hgt is not a particularly flexible format. GDAL can read it, but also says
The driver does support creating new files, but the input data must be exactly formatted as a SRTM-3 or SRTM-1 cell. That is the size, and bounds must be appropriate for a cell.
I was not able to find more details in a couple minutes of searching, but I can dig deeper. It sounds, however, like it may not be possible to cram modern, high-resolution digital terrain models into this ancient format.
I suppose the next best thing (quite a distant second!) is to generate color relief maps from the geotiffs and use those as map overlays. The problem is that in any given map view, I want to distinguish a total surface relief of only a few meters. So, e.g. 200 meters would be at one extreme end of the scale, and 205 meters would be at the other extreme. If I move to another place on the map where the elevation range is 300-305 meters, then my pre-generated, color-coded relief map is useless...