It's over 10 years ago that I had to do such operations regularly with rather unreliable networks to Southeast Asia and/or SD cards, so calculating the checksum every time on the fly was important.

Instead of the "fancy" syntax I used

   mkfifo /tmp/cksum
   sha1sum /tmp/cksum &
   some_reader | tee /tmp/cksum | some_writer

Of course under the conditions mentioned throughputs were moderate compared to what was discussed above. So I don't know how it would perform with a more performant source and target. But the important thing is that you need to pass the data through the slow endpoint only once.

Disclaimer: From memory and untested now. Not.at the keyboard.