watch -n1 $() does not run $() every interval

It likely does run every interval - but it's overwriting the same value every time because the arithmetic expression is evaluated by your interactive shell before it is passed to the watch command. You can see this if you run pgrep from another terminal:

watch -n1 echo $(( $(date +%s -d "sun") - $( date +%s ) )) 

then

$ pgrep -af watch 81 watchdogd 29311 watch -n1 echo 111748 

You can prevent early evaluation by single-quoting the expression:

watch -n1 echo '$(( $(date +%s -d "sun") - $( date +%s ) ))' 

giving

$ pgrep -af watch 81 watchdogd 29543 watch -n1 echo $(( $(date +%s -d "sun") - $( date +%s ) )) 

which will pass the expression to a /bin/sh shell by default.