The simplest solution, if you can modify /etc/sudoers, is to unset the `tty_tickets` option for your user:
Defaults:yourlogin !tty_tickets
Then you can run a sudo command before running tmux, and sudo will update your (single) timestamp and allow the subsequent sudo commands without a password prompt (within the timestamp_timeout).
This is the pertinent option because, typically, the tty_tickets option is set, which requires a password for each tty and tmux starts a new tty.