While it is true that space time dilation can cause red/blueshift, that is a distinct from the doppler effect which is the primary effect here.
(dilation plays only a small role: without time dilation our answer going from 700nm to 350nm would be 0.5c instead of the 0.6c calculated below)
See https://mander.xyz/post/17123445 and in particular https://lemmy.blahaj.zone/comment/10334166.