Yes, you are correct in saying that the "white" metal, or die casting zinc, will probably work if the design is "stronger", operating stresses lower to below the fatigue thresholds for that material. However, having used brass, I'm not sure that the higher strength alloys of brass, such as "high strength" yellow brass, are too flexible, might also give a better fatigue life. That's the whole point with them. And brass was mentioned because the fellow was used to casting with it. If bronze works, so be it, although, according to my Mark's and Materials Engineering Handbooks, bronze wouldn't necessarily be a better choice, according to the allowables listed, as above, but can't argue with whatever works.
As you suggest, I would first try the correct alloy of the zinc "pot" or "white" metal with modified designs in certain areas, mainly the base of the forks, and also the "H" section of the handle half way from the pull to the forks. Simply put, they need more x-sectional area and higher moment of inertia to reduce the tensile fatigue stresses in bending.
Making some of these for a decent price would be a boon for us.
The outside door handle is considerable trickier because of the little cantilevered arm which does the pulling. Makes a tricky casting, but I can see how the outer part which rips out in that area can be strengthen, however, in a new casting.