There's no necessary implication that "I don't mind" means you want to do the task.
If a couple had the following conversation:
Wife - Honey, do you mind changing the baby's diaper?
Husband - No, I don't mind.
One could hardly assume that the husband likes changing diapers, only that he's willing to do it because he knows it must be done.
There's also no requirement that the answer include either "I don't mind" or "I do mind"...
If he wanted to tease his wife, he might add any number of variations:
Again... ugh, I'll take care of it.
Do I have to?
Of course I mind... but I'll do it just for you.
There's tons of options and there's no reason to limit your response. If you want to say that you do mind but you'll do it any way, that's OK... though it's probably best done in a light-hearted manner.
Remember that "Do you mind" is a polite or indirect way of asking someone to do something, not always a question asking whether they actually mind or not.