I've lived in Dubai and crisscrossed the middle east on business. You are right, Dubai is the obvious choice and that's where I was based most of the time. It would have helped if you had mentioned more details about the type of business you are into.
Dubai is a little expensive but unparalleled in certain aspects. People love to live there. It's full of talent and opportunities but also very competitive. Bahrain too is comparable in some ways and there is a lot of local talent here and it's cheaper to operate a business here.
But there real answers will depend on what is your product? Will you need many local hires? Will you need large offices? Will you need to fly in and out often? Will you be dependent on hiring expat employees?
Book a call with me and I'll help you with basic information.
Just outside of Dubai is a business zone that is designed for businesses to be housed in the UAE yet be international. Jebel free Dubai I think it's called. Don't quote me on that but I would start poking around with others to get more specifics on the exact name.
My advice is to open your business office in Doha (Qatar) if there is no clients as of now in middle east. This city and suburbs is undergoing a major facelift as part of the preparation for 2022 Olympics and hence there is lot more business opportunities than any other country in the Middle East. Also, note that Qatar is located at the center of GCC countries.