Is there a word for being excellent at something you hate doing?

That's a great question and I can't think of one, but I can coin one: antagonist savant.

Antagonist defined as "A person who actively opposes or is hostile to someone or something" (https://www.lexico.com/definition/antagonist) plus the savant in idiot savant https://www.lexico.com/definition/idiot_savant

There's anti-hero:

A main character in a dramatic or narrative work who is characterized by a lack of traditional heroic qualities, such as idealism or courage.

I could easily see an anti-hero being excellent at fighting villains, saving the world, and so on, while at the same time hating it.

Some examples from popular culture:

• Taylor Durden from “Fight Club”
• Captain Jack Sparrow from “Pirates of the Caribbean”
• Don Draper from “Mad Men”
• Gregory House from “House”
• Walter White from “Breaking Bad”

"embittering expertise", is a phrase that conveys the concept of being excellent at something you hate. Here's your example completed with the phrase:

Dave's skill at placing epidurals is only exceeded by his hatred of placing them. It's a great example of embittering expertise.

"embitter" is defined by Lexico.com as

Make (someone) feel bitter or resentful.

The very thing Dave excels at (his expertise) is what embitters him, so it's fitting to describe his situation as one of embittering expertise.