Believing morality is subjective is a very dangerous and harmful contention to have, I agree, but believing it's objective is short-sighted. The more someone understands their nature the more they will understand the universal laws of morality.
No one likes being angry, to say otherwise is to repudiate the meaning of anger altogether. The "subjectivist" will argue that not everyone gets angry over the same things, but the wise understand that everyone gets angry the same way- where attachment to views leads to aversion towards disagreement and/or non-understanding.
A whole group of people may be angry about different things, but none of them are fundamentally for different reasons (fundamentally is the key word here). Where there is anger, it can always be traced to non-understanding and/or disagreement regarding views they are attached to. In order to rebut this, one must be able to demonstrate where there is anger arisen from agreement and understanding.
You objectively approach anger by practicing detachment to your views. To do otherwise, is to assure anger in your future, which is the work of fools. For the wise realize when one is free from being subject to anger (through detachment), there is happiness and goodness that can't be shaken in any circumstance, and then it can be shared. And happiness is never lost or reduced when shared, the flame of one candle is not reduced by lighting another.
This is just one example of how the Tathagata taught supreme, objective, universal, goodness and happiness.
I personally think objective morality is fairness and whats fair is objectively good as the purpose of a morality is to give us the best chance of enjoying life and not suffering. No one wants to suffer and everyone wants to be happy so it is beneficial for everybody to have rules based on fairness so no one takes advantage of someone else harming them for their gain without being punished to be fair and to prevent things that will ruin the quality of life for as many as we can punishing as many people who act unfairly as we can to prevent more unfairness and or gratuitous suffering, and unhappiness. But if thats all wrong and is only based on my opinion and all other moralities are based on opinions then there are no moral truths and morals only hold you back as you would be playing by rules and others wouldnt except for when its convenient for them. Subjective morality isnt real as its not based on anything real or objective so if there is no objective moral truths then why would you want to play by any rules at all as rules will just hold you back for no reason as they dont accomplish what is good as if righteousness is subjective then goodness is subjective as goodness is righteousness. So are morals based on something in reality making them objective or are they based on opinions?
Throughout our history we have always been evolving what we see has moral. There are lots of examples of what we did in the past, that now are seen as acts of evil, like having relations with children. Even to the extent that plays written not that long ago time wise need to be changed for today's audience.
I believe morality comes from empathy, something not just humans have evolved. We have the ability to look at someone, see something and think "I would not what that to happen to me". These feelings evolved to encourage cooperation between animals not just humans. We also have people in the human race that do not share these feelings, which we scientifically can spot with the clever new understanding of genetic markers. Humans are progressing and evolving empathy to create rules and laws, we look at new things that hurt people and we make them illegal, which is just an extension of morality. We don't always get it right but its clearly an evolution, not a set of rules that existed beforehand.