For those of you who are wondering, what different mordants can do to the dye, here is sappan wood with alum (red) and the other half of the same dye with iron mordant (purple-grey-black).
Besides changing the colour (or tone) itself, in some cases mordants used of fabric also improve, sometimes dramatically, light- and wash-fastness of the dye. I don’t know what will happen with eggs, I guess, the time will tell.
Here’s the iron mordant I got and used:
Technically speaking, you could just use the edible grade iron sulphate supplement from a drug store, it’s the same formula most of the time.
Another proof that not all that works for fabrics works for eggs also: the iron added to coreopsis dye did nothing, in fact, it just killed the dye, even though iron seems to work well with coreopsis for both plant and animal based fibres.