"With pleasure" and "It's my pleasure" both sound a bit excessive for the chore, and mean more than "I'm willing to." They mean that I would be delighted to look after little Pepe and feed him three times a day and walk him and clean up after him and replace the various things he'll destroy while you're gone. But perhaps you love the dog that much.
If you're enthusiastic, you might consider, "I'd be happy to."
I must admit to greatly preferring "With pleasure" to "It's my pleasure", which sounds extremely pompous and self-important. I couldn't say it.
It may be that I am influenced against "It's my pleasure" by the fact that I was once alone, quietly sitting minding my own business in our Common Room at work, late one cold winter Friday evening, when a very elderly and very reactionary and, as I thought, very stuffy colleague came in and told me a story about a Geordie who went into a hat shop.
"Good Afternoon, Sir, and what is your pleasure?" asked the smart assistant.
The Geordie replied, "My pleasures are fucking... and pidgins, but, right now, what I want is a flat hat."
I think the Geordie was right. We shouldn't talk like that about things being our pleasure.