Likely not... that's not how the refillable Dining Plan mugs are supposed to work.
If you are trying to cut back on paper cups, then you could buy any souvenir mug and pour it in there. But if you're trying to get an "extra" off the menu by pre-buying a drink, no, it doesn't work like that.
I think the only "wiggle" is, if you HAVE Dining Plan, and you have your mug, because your Dining Plan meal comes with a drink, you can grab a to-go beverage from the case. Milk, water, juice, soda... We were always refilling our mugs and stocking up the Park backpack with the bottled beverages each morning at the food court.