It looks like the answer is no,
The biggest problem you face in dyeing a coat is that it probably has a DWR (Durable Water Repellent) finish. This is a coating that is applied to the outside of the fabric to resist water. It's very important to have a DWR finish on a down coat, because down loses all insulative ability once it gets wet.
If your coat has a DWR coating, you will not be able to dye it successfully, and you probably won't be able to paint it satisfactorily, either. Occasionally one sees advice from non-experts to ruin the DWR finish with fabric softener in order to dye it; unfortunately, the DWR finish will still be there enough to prevent adequate dyeing, and fabric softener itself causes problems in dyeing.
It doesn't look like the dyeing process is much fun either,
I'd rather not subject a down coat to dyeing, since you have to simmer the coat with the acid dye, plus an acid such as vinegar, for some time, stirring constantly. I doubt that you have a large enough cooking pot to allow the coat to move freely, and if you do, you're unlikely to want to spoil it by using it to dye. You should never plan to reuse a dyeing pot for food later on; textile dyes, including Rit dye, are not safe to use in food preparation containers. Also, the vinegar will react badly with aluminum, so you cannot use an inexpensive pot; you'd want a stainless steel or enamel pot. I don't recommend dye for your nylon coat.
It looks like you are out of luck.