You require a login shell. The tmux(1) man page discusses login shells in four places. You should test them all. You might start with the option I've highlighted, since the man page says it is necessary.
Execute shell-command using the default shell. If
necessary, the tmux server will be started to retrieve the
default-shell option. This option is for compatibility
with sh(1) when tmux is used as a login shell.
-l Behave as a login shell. This flag currently has no effect
and is for compatibility with other shells when using tmux
as a login shell.
Set the command used for new windows (if not specified
when the window is created) to shell-command, which may
be any sh(1) command. The default is an empty string,
which instructs tmux to create a login shell using the
value of the default-shell option.
Specify the default shell. This is used as the login
shell for new windows when the default-command option is
set to empty, and must be the full path of the
executable. When started tmux tries to set a default
value from the first suitable of the SHELL environment
variable, the shell returned by getpwuid(3), or /bin/sh.
This option should be configured when tmux is used as a