Hi buttercupbetty,
There are some very good suggestions here but I would also consider looking at the basics to see which materials you would be comfortable having in your mattress even if they aren’t ideal. The options you have (and some may be more difficult to find or may be in higher budget ranges) include …
Polyfoam: CertiPur certification would mean that there is little chance of offgassing and VOC’s that are harmful by most people’s standards although some people may not be comfortable with them or be more sensitive to VOC’s than the majority of the population. Most people who are OK with polyfoam in other furniture and household products would also be OK with them in their mattresses. there is also more about the two main chemicals used to make polyurethane foam (polyfoam) in post #2 here.
Memory Foam: Memory foam is a type of polyurethane foam with more chemicals added to give it a slow response and additional density so the previous comments about polyfoam would also apply although memory foam seems to affect some people more than polyfoam regardless of the certification.
Latex: From your tests it appears that you don’t have a type I allergy so it may just be a matter of having someone put the mattress together for you so you have no contact with the latex. You would have a choice here between 100% natural Talalay (which you appeared to react to but which may be OK if there is no contact), blended Talalay, 100% natural Dunlop, or blended or synthetic Dunlop. There are also some very soft continuous pour Dunlop versions available in either synthetic or 100% natural. All the latex you are likely to encounter of any type will have been tested for harmful substances and VOC’s either by Oeko-Tex or Eco-Iinstitut (which are more stringent tests than CertiPur) and would be “safe” by most people’s standards.
Natural fibers: This would include cotton, wool, and horsehair (which are generally found only in much more costly mattresses) along with feathers, down, and silk or any other natural plant or animal fibers. Firmness may be an issue here but this would vary with the specifics of the product and how it was made.
Rubberized Coir: Coir is a natural fiber that comes from coconut husks which would be “safe” and can sometimes be certified organic. Rubberized Coir is coated with latex to give it more spring and resilience and to help with water resistance. This is sometimes used as a firm support layer in a mattress or as an insulator material over innersprings.
Synthetic fibers: These are generally various types of polyester or Dacron fibers that are synthetic and usually not as durable as natural fibers but they can be fairly soft and non toxic. They are generally used in quilting layers or as a “down alternative” product in a “fiberbed” or topper.
Innersprings/Steel: This would be very unlikely to cause any issues. There are also microcoils that are used in comfort layers but they are generally used in combination with thin layers of foam above them. They come in a wide range of firmness levels but would also have some kind of padding over them. There are a few people who may have some concern with electromagnetic fields that they believe are generated by the coils in an innerspring (see the end of post #3 here).
PET densified fibers: There are some types of materials that use recycled PET fibers which have very little offgassing (see an example here and a manufacturer that uses it here) but these are synthetic materials and I don’t know the softness levels they have available.
Buckling Column Gel: These are also more costly and generally used with a layer of some type of foam above them but they are very pressure relieving and durable and “safe” in terms of VOC’s and offgassing.
Soft Solid gel materials: This is similar to the gel that is used for buckling column gel or added to other types of foam materials. It is very safe and has little to no offgassing but it is generally only used in thin layers because it can be quite firm (see post #4 here). An example of this would be Technogel but these is only used in combination with other components and foam materials and not as all the layers of a mattress.
Waterbed bladders: These are generally vinyl so you would need to know how comfortable you are with this. They can be soft sided or hard sided and used as just a water bladder or with additional comfort layers on top (usually some kind of foam). There are also some people that are concerned with the use of pthalates in any water or air bladders that use PVC materials (see post #2 here).
Airbed bladders: These are generally made from either vinyl (see the comments under waterbeds) or from rubber or in some cases urethane and also use various types of foam padding over them because it would be uncomfortable for most people to sleep directly on an air bladder.
You may also want to try testing some of the “ultra premium” innerspring / natural fiber mattresses listed at the end of post #4 here as a reference point just to see how soft a mattress that uses natural fibers can feel for you.
These are most of the materials and components you will encounter and once you have put together a list of what you would be comfortable or “OK” with using in a mattress (even if it’s not your perfect ideal) then you will be in a better position to know which direction to look.
Phoenix