Why is my function not re-evaluated in PS1?

According to Bash prompt Howto:

[21:58:33][giles@nikola:~]$ PS1="[\$(date +%H%M)][\u@\h:\w]\$ "
[2159][giles@nikola:~]$ ls
bin   mail
[2200][giles@nikola:~]$

It's important to notice the backslash before the dollar sign of the command substitution. Without it, the external command is executed exactly once: when the PS1 string is read into the environment.

When you used $(..) in double-quotes, the shell evaluated the command substitution before assigning to PS1. Thus, PS1 contained only the output, not the command substitution itself. Instead, either use single-quotes, or escape the $, so that the string is passed as-is to PS1, and then evaluated when the prompt is set:

$ PS1='$(pwd) $ '
/tmp $ cd /var
/var $ echo "$PS1"
$(pwd) $ 

Compare:

/var $ PS1="$(pwd) $ "
/var $ echo "$PS1"
a /var $  a
/var $