Hi litesleeper,
In the list for Charlotte, the odds are that if anyone would rebuild the mattress it would be Dilworth. There are many manufacturers however that stay away from this because it can also create issues (both real and perception) for the manufacturer that they would rather not deal with. The real issues are connected to having used materials mixed in with new materials (such as bedbugs) that could affect their manufacturing and the perceptual issues is that they could be seen as selling mattresses with used materials and aren’t labeling them as such even if they are only doing this on request for a specific individual.
In any case … it may be worth a call to find out if they can or will do this.
The other “rebuilding” alternative is what you are suggesting and is often called “mattress surgery”. This would involve taking off the ticking/cover (removing the tape edge with a thread cutter), removing the low quality foam and as you suggest replacing it with a higher quality comfort layer (assuming the core is still good and not impressed) and then buying a new zip cover (and there are good sources for this where you can buy zip covers of many different types and quality levels) that was the correct thickness to enclose your new mattress. some sources for this type of DIY project are listed in post #4 here.
One of the challenges you may face with this is if the layers are glued then it can be difficult (but not impossible) to remove the lower quality foam layers and this needs to be done very carefully. You would need to carefully “cut” the tightly glued areas off the core as you gently lifted the comfort layers off.
Your core is a lower quality latex that has been fabricated to provide “zoning” in certain areas with “inserts” in the latex but the odds are good that it would still be in good condition. While blended “mostly synthetic” Dunlop latex is not as good a material as all natural or even a 50/50 blend, … it is still better than other foam choices.
“Mattress Surgery” can be a fun (and challenging) because it still involves a bit of research into the layers to put on top of the support core but it can also be satisfying if you get it right and it can certainly cost less than a whole new mattress. In most cases … it’s the comfort layers (the softer foam in the upper padding and quilting of the mattress) not the support layers (like an innerspring, latex or even polyfoam support layers) of a mattress that are the “culprit” of mattress softening issues and the loss of comfort and support. If you add high quality latex to your core and use a high quality cover … you could end up with a very good mattress that would cost you several thousand dollars in a store.
So if there is nobody locally who can do this for you and if the other lower cost “solutions” don’t work to the degree you want … then this could be an alternative if you are going to replace the mattress anyway.
I sleep on an all latex two sided latex mattress with a 4" middle support core that is “soft” as far as mattress support cores go but is rated as a “medium”. On either side of this is 3" of soft latex with a quilting that includes .4" of quiltable latex and a down substitute material. The ticking is a Belgian Jacquard. It is very unlikely that this particular layering would work for many people and it’s not a layering I would recommend at all for the majority of people … but I specifically designed it and had it built based on some very careful testing that took into account some unusual preferences. It was designed for my DH with my own needs and preferences as a secondary issue (although it is fine for me as well). Your own experience with back issues on your mattress would indicate that the odds of this type of layering working for you are very low.
Hope this helps 
Phoenix