So mentioning it's a fatigue issue related to my illness would do the trick? ...
Calling it a "fatigue issue" will get you the recommendation of using a wheelchair or ECV. As others have posted, just because you don't use one in everyday life does not mean it won't be tremendously helpful to you at WDW. The average person walks 6+ miles
per day in the parks, I doubt you do this much walking day-after-day at home. People have reported that by using a DAS they actually end up walking more because they back-track more.
Most queues at WDW are indoors and air-conditioned. You will find much more exposure to heat and humidity outside -- between rides and while waiting for a DAS return time. You might consider some cooling techniques -- Frogg Toggs cooling scarves, misting fan, cooling vest.
You certainly may inquire about a DAS and discuss your needs with Guest Relations. However, I'd be prepared with back-up plans such as the suggestions others have given already, in the event a DAS is denied. Use of FP+ and a good touring plan can be quite beneficial to getting through the parks with shorter waits. Accommodation does not have to meet a specific person's preference, just adequate for the situation.
... I'm sorry but I just don't see how this is a better system. ...
"Better" for any system is in the eyes of the beholder. To WDW, this is a better system that manages crowd control to a greater extent than the GAC. Many posters have indicated the DAS combined with FP+ has been a better solution for their family, allowing them to accomplish several rides in a short period of time. There are those guests for whom the DAS system may not be ideal -- there is no one single solution that will accommodate all needs. There are a variety of disability needs not accommodated by the DAS but handled more appropriately by other methods in the parks.
.. I have a disability - handicap permit and all - recognized by the government as a disability ...
All disabilities are not the same, as such there is no one-size-fits-all accommodation. To get a handicapped parking permit you had to submit medical documentation of need for close parking - something not available to the general public. The goal of DAS is to create a waiting system equal to that experienced by the general public, not a preferential experience. To get a DAS you have to explain why you cannot wait in a standard line queue and why other accommodation suggestions are not adequate.
Consider your symptoms. Consider what could be the detrimental impact of waiting in lines. Think about alternative ways to accommodate or alleviate or avoid that symptom/problem. If there are no other options, the DAS may be your best option.
Enjoy your vacation!