Hi latexintheatx,
There are faster responding memory foams available (foam chemists are always coming up with new versions), but if you’re expecting bounciness, then you generally wouldn’t be looking at memory foam, unless it was a small amount and combined with other more resilient materials in the mattress. Obviously this would be something you could feel if you test items locally, but if you shop online you’d want to speak with an informed consultant and ask them if they have any information on how the recovery rate of their memory foam compares to older Tempurpedic models to assist you with your selection.
If there are small particles or latex dust in the latex from cutting that are connected to the rhinitis then a cover could possibly help, and I’ve also seen a number of instances where a dust mite cover with small micron sized pores solved similar issues as well. There is a little more about allergy encasements and pore sizes in post #2 here.
The membranes wouldn’t protect against any offgassing (they are vapor permeable) and to protect against this you would need a clear polyethylene plastic cover in the range of 5 mil or thicker (clear polyethylene is non toxic) but in your case it’s the particles that are probably most important and the polyethylene would not be breathable at all and would likely cause temperature regulation issues as well as trapping any moisture in the mattress.
Regardless of the cause (whether from latex or some other component within the mattress) it still might be a good idea for your husband to be tested by a health care professional, whether or not you keep the mattress.
I’m also sorry to hear about your father-in-law. 
Phoenix