Can I add food coloring or color dye to the cooling system? (UV dye won't help my situation)

Context: My 2004 Dodge Neon’s transmission cooler failed, causing the transmission fluid and antifreeze to intermix in both systems.

I replaced the radiator, thoroughly flushed the cooling system, and have now changed the transmission fluid 4 times to get all the coolant out of it.

However, my transmission fluid still looks somewhat milky! And I’m trying to diagnose it. Maybe I’m still getting all the old coolant out and there’s nothing to worry about. Or maybe the new radiator has a defective transmission cooler. Or maybe water’s leaking into the automatic transmission somehow.

So I’d really like to put a bunch of food coloring or something in my coolant…make it dark green, or something…so that I can rule out a defective transmission cooler.

I see that UV dye exists for cooling systems, but I’m skeptical its UV properties would be visible diluted in a gallon of transmission fluid. (Not to mention it’s a pain emptying the transmission pan.) Whereas if I could just turn the coolant dark green or something, I could just use the transmission dipstick to check the transmission fluid’s color.

Is it safe to put food coloring in the cooling system? (How much?) Or, is there some other way to color the coolant that would be detectable when diluted in the transmission fluid?