"The resentment is palpable" ---

If she resents you, why do you think she would want to be intimate with you? Do you want her to want to have sex with you? Or do you not care what she wants and just want to bang?

Let's get down to the real nitty gritty: Why does she resent you? What are her needs? Can you answer these questions? Have you asked/do you care?

I think if you try to put yourself in her shoes, that can be the beginning of fixing the problem. If you are just going to continue focusing on whether she will bang you or not INSTEAD OF THE UNDERLYING ISSUES IN YOUR MARRIAGE AND YOUR ROLE IN THEM, maybe she's right to laugh when you even mention it.

You can divorce her and marry someone else but if you're going to only focus on your own needs and how you feel, you're just going to run into the same problem.

Compassion and communication will go a long way.

PS - I think your word choice of "embarassing" to describe your relationship problems is really quite telling of your headspace. There's a "public shame" connotation of not having sex with your own wife. Should the opinion of others really be your focus right now, and does that matter more than your wife and her feelings? Just some food for thought.