We had this 2yrs ago except it was DH's best mate, they had known each since primary school and his brother's were also going to be going. There was no way I was not going to let DH go. It was just a matter of deciding if I would go with DD (4mths at the time). In the end he went for 7days.
He had a good time and the kids loved imagining what surprises he was going to bring home.

I would either go with Tinks idea or go yourself for minimum amount of time. It's one of those life opportunities you may regret later.
Chile would be a fascinating place and I don't think you would be bored at all.