Close. But 드시게 is not correct in that place because its an honorific form given to yourself when you're trying to honor the other person. And I'd rather use 감히 instead of 겸손히, which can be roughly translated into "dare to"

"존경하는 각하, 맥주 한 잔만 마시게 해달라고 감히 부탁드려도 될까요?"

Another Korean here. Usually in Korea, one could refer to an unknown someone as "선생님(teacher)". "각하", on the other hand, has been referred to the President of Korea. So it really sounds weird to a Korean. When someone calls you "각하". If I were to be a customer, then I would be called "고객님". Where "고객" means customer and "님" is like "sama" in Japanese, and "sir" in English.