With tafwid talaq does the wife have to repay the mahr?

No, because she's not the one that is initiating the talaaq.

She's initiating it on the behalf of the husband who gave her the permission to.

So, it's as if he said go to the store and buy me a Red Bull.

She doesn't drink Red Bull. She doesn't want it.

But he's asking her to do it for him.

So, she goes and uses his money and buys it.

Yes, she's the one that procedurally bought it, but she bought it for someone else. And the benefit of someone else and the responsibility of someone else.

So, similarly and I'm not trying to say that talaaq like Red Bull, but I'm just saying that she's essentially wakil, an agent on his behalf.

She's working with a power of attorney to do something that legally is his right to do.