Thank you for the clarification
If you're not in Dubai, registering at Amphur seems like a more direct solution; quick visit to get a marital affidavit at US Embassy, translate, get MOFA stamp then head to Amphur.
In my case, I got a bit of an issue with my domicile's Amphur in Bangkhen District, BKK, they wanted me to bring all the documents to them to verify before I can get an appointment (claiming that there're lots of forgery around lately), they said they will hold on to the documents for a few weeks. My wife insisted that we register marriage on 25th Dec (!), so I had to find other Amphur to make it happen. Ended up registering married in a faraway District of Bang Sue who allowed me to do urgent appointment at the time.
I believe each Amphur has their own way of doing things, so you might wanna check up on them beforehand.
Also, you may check for any certification service in UAE as well, I once hired an agent there to handle similar kind of attestations as you mentioned; took about a few weeks. If the cost ain't bad, it is a viable option.