Thanks everybody for the ideas. I decided to suck it up in the back of the plane and save the money for a personal trip in the future.

I wasn't aware of the EK Arabian Airpass and definitely will use that option to fly DXB-MCT rather than deal with the SHJ airport and a different airline. The pass is a great deal and I wish I had more vacation time available post-business trip to explore some of the places it can take you. Perhaps next time!