Sorry for the late reply. I've been ill.
I projected my DEMs into WGS1984 (not UTM) and made sure my mosaic of SRTM 1 Sec DEMs covered my entire radar image. The resultant mosaic actually covered a lot more than my radar image. Sadly, it still did not work. I plan to try clipping my DEM to match my radar image, see if maybe that helps.
I know that the external DEM is ignored as the product produced when I select an external DEM is identical to the product produced by the SRTM 1 Sec autodownload option. The result only covers a tiny portion of my radar image and does not match the extents of any of the 1 Sec tiles or the mosaics I've created and told SNAP to use.
My location is the Erta Ale volcano near the Red Sea if that means anything.