Out of the two solutions, you can probably work on this one, where through the bootstrap 3's inserted.bs.tooltip
event you can customize the tooltip.
$(document).on('inserted.bs.tooltip', function(e) {
var tooltip = $(e.target).data('bs.tooltip');
tooltip.$tip.find('.tooltip-inner').addClass($(e.target).data('tooltip-class'));
});
the .find
part is relevant, since it targets the proper element.
fiddle: https://jsfiddle.net/eqsLp8gt/2/
EDIT:
Updated fiddle, suggested by the OP in order to work with the arrow styling as well:
https://jsfiddle.net/eqsLp8gt/3/