A safe chemical dye would be (Potassium Permanganate) it is in a dry powder form, mix it with enough water to get it liquid and it becomes a perfect "tie-dye". I got mine from a local Pharmacy.
Since you are looking for only a quart, may I assume you need it for modeling? It will work for coloring, but it does not protect once dried.
It will also hold up outside and even under water. Once mixed with water and dried it becomes completely inert in a day or two, consequently it cannot be stored once mixed.
IMPORTANT>>> Mix by adding powder to water not water to powder, it will create heat when mixing. It is used as a cure for various fish parasites and is harmless to all but the parasites. It is a very deep purple when mixed.
Just soak the ties watch for your favorite color of blackish brown and dry. Warning... Use gloves! this stuff will stain your hands and clothing so you look like you have been in a tannery. And it will need to WEAR off.
Norm'