walnuts and other botanicals have been used for centuries to dye clothing. I tried dying a canvas coat once and found most of the color washed out. How do you "set" the color . I guess thats the correct word for keeping the color in the cloth.
just

Ad the walnut die to your last rinse cycle.

Call it a walnut stain rather than a die. Stains are really hard to get out.

There is no mordant for walnut dye. I try to minimize washing my walnut dyed items.

When dyeing traps is rust (oxide) the "mordant" for the walnut tannins?

I'm not sure the concept of "mordant" applies to dyeing traps. In any case, whether it be walnut or logwood, rust does help steel take and hold the dye.

Adding rust to your walnut hulls will certainly give you a much darker color. Whether it will hold up to washing better, I'm not sure. When I tanned hides with sumac, I would often boil traps in it to get an almost black color, then used it for hide tanning. Sumac leaves normally make a pretty light color.