Hi anotherscott,
Welcome to the Mattress Forum! 
The first thing I would do is have some testing done by your partner to see if they indeed have a sensitivity to latex. There is a wealth of information about latex allergies in this thread and the articles it links to. Overall there are three different types of sensitivities/allergies to latex, most of which are due to contact with the type of latex used in medical equipment, and since most of these proteins are rinsed out of foamed rubber latex, it is rarely an issue (you’re not in direct contact with latex when sleeping upon a latex mattress), but I understand if someone has a concern.
If you’re eliminating latex and memory foam, your other options would generally be innerspring, polyfoam or natural fibers. You could create your own DIY mattress using components of your choosing. Some of the online foam mattress manufacturers will split each side upon request. You’re looking for something quite specific, and I don’t track the individual variations and specifications of all of the manufacturers and companies mentioned here on the forum, as it would be a larger task than anyone could maintain with a market that constantly changes, but two good places for your to begin your search for mattresses that use no latex or memory foam but are configurable would be the site members here, and the listing of some of the simplified choice mattresses here. If you find a product that looks interesting as far as componentry but you’re not sure about the ability of it to be split, I would be sure to contact them to see if that is a possibility, as it sometimes can be accommodated. There’s also a listing of mattresses that use no foam here.
Phoenix