You are actually not allowed to bring your own mug. This is from the FAW section of the website:
Should we bring large insulated mugs?
Drinks are to be served in glasses or cups provided by Couples Resorts (Management on October 12, 2004 restated this policy). The glasses at the bars are standard size that you would get in the States.
However, you can bring a water bottle or mug to use for ice water on the beach. No worries, mon! Your drink won't ever run dry from the bar!
We bring a couple of small size (12 oz.) insulated mugs to use on the beach. While the bar tenders will not mix or serve a drink in your mug, there is no problem with pouring it from their glass into your mug after being served. Keeps those pina coladas nice and cold.