Update (new answer):
dtach has a
-z option with the description "Disable processing of the suspend key". Confusing if you ask me, but its effect seems to be that the Ctrl+Z is passed through to Vim instead of being caught by
More general answer: a program like
dvtm has pretty much absolute control over what gets passed through to whatever's running "inside" of it. It's like having a secretary take dictation -- you can say what you like, but the secretary controls what actually appears on the paper. So if you want the end program to receive the Ctrl+Z, you have to get all of the middle layers to cooperate, whether that be through command line options or source code editing.
Looks to me like this is a bug in
dvtm. From inspection of the source code, it's not catching the
SIGTSTP signal, which is what is sent to your terminal's foreground application when you press Ctrl+z. Since it's not catching the signal explicitly, it falls back on default behavior and gets suspended. What
dvtm needs to do is catch this signal and pass it along to one of its windows.