I guess my argument for objective morality would go like this:
First, establish objective reality. If your audience doesn't accept that, then there is no reason to continue.
Then, I'd do the "argument from the hamster":
If you want to keep a hamster alive and thriving, you have to follow certain rules.
The same thing is true if it's a human instead of a hamster, although the rules are more complex. (Humans don't thrive in cages.)
The same thing is true if the human you are trying to keep alive and thriving is yourself.
That argument should be sufficient to demonstrate that an objective morality exists. It doesn't say what the rules are, but that can be the next step.
p.s. I'm aware that this argument takes it for granted that the purpose of morality is to keep yourself alive and thriving. It's possible to explain why that is the proper purpose, but I'm not doing it in this post.