Command substitution never affects the value of $?, and that's the only way to execute anything from within the prompt string.
I don't think it's possible to cause the expansion of PS1 to alter the value of $?. If you (or anyone else) knows of a way, I'd appreciate knowing about it.
Since you neglected to be specific, I'm assuming this is for bash.
It seems like what you need is a way to detect when no command was entered at the prompt. Bash makes available the command number through the escape sequence \#. This value (at least with the version with which I tested) is not incremented for a null command. Testing its current value against its value at the time of the previous prompt's expansion should be sufficient to make a determination.
Since any variables set in the command substition subshell are lost when the shell exists, and since environmental changes cannot propagate from a child to a parent, the only way to store the command number for future inspection is to write it to a file.
The following example is intended only as a proof of concept. Aside from the minor inefficiency of reading a file for each prompt, it doesn't support simultaneously interactive shells (although it could, with some work and help from $$). The prompt consists simply of the exit status followed by a colon and a space. If no command was entered, regardless of the value of $?, a 0 is printed (analgous to not printing the X in your original problem statement):