Software :: GIS - Crop Then Reproject A Raster DEM?

I have the ETOPO1 global relief model, [URL]...in georeferenced TIFF format. What I want to do is crop out a section (specifically the area of New Zealand's continental shelf), and then reproject that section into a different projection (the New Zealand Continental Shelf Lambert Conformal 2000 (NZCS2000) projection).

Matters are complicated somewhat by the fact that the region in question crosses the 180 degree longitude line. I've tried googling, reading the GRASS manual, playing around in QGIS, and even have the book "Desktop GIS" by Gary E. Sherman. But the seemingly simple task of cropping a raster DEM has defeated me. I cannot do it in ordinary imaging software, because the bits per pixel of the raster is too high; programs like the GIMP would grossly quantise the relief.