Mattress “safety” is a very complex subject and each person can have a very different answer to the question of “how safe is safe enough for me?”. Some people are sensitive to some materials and not to others so the answer can be unique to each person.
Post #2 here and the other posts and sources of information that it leads to would be a good place to start your research but in very general terms most people that are looking for the “safest” materials would do best with materials and components such as innersprings, natural fibers, or different types of latex in a combination that works well for you in terms of PPP. In terms of the different types of foam … memory foam is usually the least well tolerated by people who are very sensitive to chemicals (such as those with MCS or multiple chemical sensitivities) followed by polyfoam with latex usually being the best tolerated of the three but there are also individual exceptions to any “general” suggestion. Some manufacturers or retailers will also send you samples of the materials or fabrics they use in their mattresses so you can test them in person before making a purchase.
In the case of latex … while most people tend to lean towards 100% natural latex (or latex that has an organic certification) in circumstances like yours and feel more comfortable with more natural or organic choices … all the latex you are likely to encounter (either Dunlop or Talalay in 100% natural versions, synthetic versions, or a blend of the two) have been certified by testing protocols such as Oeko-Tex or Eco-Institut for harmful substances and VOC’s. In general the only real objective measure of the safety of materials is the certifications they have.
Fire barriers that are made either from wool or a rayon/silica inherent fabric (or a mattress that has no fire barrier at all) would also be “safe” choices.
The first link you listed appears to use mostly organic materials but it only has 2" of latex and the rest of the 9" appears to be natural fibers including organic cotton and “virgin wool” (I don’t know how the wool has been processed) so while it would certainly be a “safe” choice for most people, your own testing or experience will be an important part of knowing whether it’s a good match for you in terms of PPP (Posture and alignment, Pressure relief, and Personal preferences).
The Total latex mattress you linked includes your choice of either Blended Talalay or 100% natural Dunlop for the support core and a blended Talalay comfort layer in your choice of firmness level (which can also be exchanged for a different firmness level after a purchase if necessary). The quilting in the cover includes organic wool and a thin layer of polyfoam which most people would be fine with unless they wanted to avoid polyfoam completely. Dreamfoam is also one of the members of this site which means I think highly of them and they compete well with the best in the industry in terms of quality, value, and service.
I would also make sure that you’ve read the tutorial post here because no matter how “safe” a mattress may be it’s just as important to make sure the design of the mattress is a good match for you in terms of PPP and your “personal value equation”.
Hopefully this can help you decide on whether any mattress you are considering is “safe enough for you” based on your history with different materials and your own educated “best judgement” and will also help you find a mattress that is suitable for your body type. sleeping positions, and personal preferences.