I have heard that where DESS and DC are concerned tenacity and brown-nosing pays off! Just don't apply and then sit back and expect to hear a yes. You have to keep badgering and badgering.
Even if you send them somewhere else as a temporary measure it's worth keeping them on the waiting list for DC if you have your heart set on it, and keep phoning up OVER and OVER - I cannot stipulate this enough! The turnover of all the schools (even the very good ones) is very high here so you have to keep on the ball. It's about lucky timing.
DESS, DC, PH and Sherborne will all want to test your children and will charge for the testing and an application. When I phoned DC they told me my son's year group was so over-subscribed that there was no point in my even asking for a visit! They'll always happily take you money for the testing though. (They charge to test and to apply.) Compass do test, but it is more of a case of getting to know your child's needs/strengths to assess whether thay can accommodate them, rather than automatically looking for someone who will keep their results high. I have heard that kids who are really struggling at DC and Sherborne as left to flounder until they eventually leave or are asked to leave.
The DC tests (and Sherborne as well I think) can be done online or you can go out and visit and have them tested there, so they can look around and get a feel for the place. Although in the end you are lucky if you get to pick and choose based on your gut feeling. I think it very much a case of if you pass you are in and if you don't then tough - someone else who can pass, or who tested with higher marks that week will have that space. It's extremely competitive at most of the highly sought after schools, with the exception of Compass who are much, much more inclusive, but as you say - very expensive.
If your DH has been posted there by a British company you get priority at DC.
You are more likely to get a place easily at Compass because the senior school is still relatively new, and it has plenty of room for expansion. I think people are often a bit reluctant to try it for seniors as it doesn't have the solid track record in Doha that DC or PH as, and of course it's expensive so people are understandably nervous. They had their first cohort of IGCSE students last summer who did fantastically well, but then it was a very small group, so it's hard to know if the results were skewed by a handful of super-brainy kids. They got 100% A* at English though.
Compass is very bottom heavy with younger children. I think as those children come through, and they open their new senior campus in two years or so (allegedly) then Compass will be a big player to rival DC and Sherborne. Sherborne lost a lot of children across many year groups towards the end of last school year, but they seem to be sorting themselves out a bit now. Most people are happy with it, but it is very pushy and results driven - I know someone who teaches there and she finds it quite disappointing as she doesn't think they nurture the 'whole' child.
My child goes to Compass and honestly, I could not be more delighted with it.
It is worth every penny in my book.