It never hurts to ask at any restaurant, and if they have what you want, you may be allowed that with your credit. However, be prepared to order something from the menu as that is what the restaurants are reimbursed for your credit.
Every restaurant has a limit on the cost of the drink you can get with a DDP, even if they don't advertise it, and it's usually somewhere around $17. That being said, sure, if you don't want one of the listed cocktails you can get a gin and tonic and you can even name the gin, provided it's not over the limit (which pretty much no gin would be. Whiskys might be another story).