I whipped up a quick implementation of Send & Archive for fun which you can hook up with something like Greasemonkey:

  O.require(['mail', 'compose'], function () {
    const composeController = FastMail.mail.screens.compose;
    const mailActions = FastMail.mail.actions;
    const mailToolbar = FastMail.views.mailToolbar;
    mailToolbar.registerView('sendArchive', new O.ButtonView({
      type: 's-send',
      icon: 'icon-paper-plane',
      _isSending: O.bind(composeController, 'instance.isSending'),
      _mayMove: O.bind(mailActions, 'mayMove'),
      isDisabled: function () {
        return this.get('_isSending') || !this.get('_mayMove');
      }.property('_isSending', '_mayMove'),
      label: O.loc('Send & Archive'),
      target: {
        doIt: function () {
          composeController.fire('send');
          mailActions.archive();
        }
      },
      method: 'doIt',
      shortcut: 'cmd-shift-enter',  // (Fairly arbitrarily chosen.)
    }));
    const config = O.clone(mailToolbar._configs.compose);
    config.left.splice(config.left.indexOf('send'), 0, 'sendArchive', '-');
    mailToolbar.registerConfig('compose', config);
  });

It’s not elegant, but it works. It’ll probably break when the web UI switches to JMAP some time next year.