As far as I know tmux already copies things straight to X11 clipboard by default (at least in version 3.5a) and no configuration aside from enabling mouse is necessary.

For me, customized Vim search is just miles ahead of everything else so I prefer that (not just for the search, but also to open files with gF, etc.).

I also have a couple binds in Vim to deal with tmux:

    vnoremap <leader>ty "ty:call system('tmux loadb -', getreg("@t"))<cr>
    nnoremap <leader>ty "tyiw:call system('tmux loadb -', getreg("@t"))<cr>
    nnoremap <leader>tp :let @t=system('tmux showb')<cr>"tP