short: no
long: The behavior is terminal-dependent...
The terminals that you would use `tmux` on hard-code their behavior to match the behavior of **`xterm`**.
While `xterm` provides a way to assign different actions to different mouse/button/modifier combinations (see Default Key Bindings), it makes assumptions about the use of the `shift` key as well. The `translations` resource is flexible in some respects, but does not (aside from the `keymap` feature introduced in X11R4) provide for changing the translations back/forth as one starts and stops mouse-mode.