Welcome to the forum! While it is a beautiful dresser, I don't think it is worth spending that much for refinishing. Not to mention, I think that estimate is ridiculous in the first place. In it's current condition, it's worth about $45.00. If refinished and in beautiful condition, about $150. I would shop around and see if you can get the refinishing done for a few hundred. Then, you'll have a great old piece of furniture for about the same price as a new dresser. Maybe a bit more than a new dresser would cost but it will be worth it for it's age, in my opinion.
You, know, now that I look at the pictures, the condition doesn't look too bad. Maybe you can find a furniture repair place that can fill/color match some of those scratches for much less than refinishing the whole thing.