I was wondering why it "must " be done after installation, and why it would be simpler this way is all. I realize it would likely be easier but I think you end up dying more or less only the exposed side of the wrap, which is fine "if" it never moves. I would use a single roll of white, unspool it, dye it, then re-spool it. This way it is all the same color. I know it's more work, but possibly less problems later.
OK, I understand that you are wondering and that's fine.
Actually, that's why we CMs hang-out here. So we can address the questions of those that wonder.
As to the "must", there is no must way of doing things in this craft.
There are ways and techniques that are acquired through experience, trial & error.
After 23+ yrs of building I've learned from experience what works for me.
Sometimes I'm even willing to share some of what I've learned.
It doesn't mean that anyone has to believe me or even pay attention.
I'm self-taught in just about everything I do, be it welding, machining or building cues.
The point being, what I've found to work best for me may not be best for anyone else.
If someone comes away from one of my posts with a bit of knowledge they didn't have before,
I'm a happy camper. That's the reward of sharing.
Next up, wraps moving.
Once installed, a linen wrap should not move. If it does, it wasn't installed correctly.
My wraps don't move until I (or someone else) takes them off.
If they move in the field, you need better glue and/or better technique.
As to the wrap moving to expose an undyed suface, not going to happen.
The dye is suspended in liquid, usually water. It satuates the cordage.
Hypothetically, if the wrap did move around, it wouldn't show any other color than what I dyed it.
Some wrap colors just aren't available and where the client gets the idea for their
preferred color scheme, I have no idea. But, they want what they want so you try to accommodate.
The method that I use was shared to me, I didn't come-up with it.
It's so slick, trick & simple that I kick myself for not thinking of it.
This was one instance where (with sharing), what goes around - comes around.
My sharing was returned to me.
I hope that you and everyone else has a GREAT holiday season.
KJ