Given the UV is breaking-down the Dometic plastic cover, I would paint the new cover. I suggest you pick a darker accent color on the rig to match. I think a darker color will protect better than white paint.
Our Dometic cover was painted silver as part of the full body paint option. They painted over the Dometic name decal but it surprisingly has been perfectly fine. It has not flaked off, very well bonded. Our rig is 8 years old now but is stored indoors. I can't see the name from the ground, only when up close on a ladder. The cover is textured plastic, the decal is much smoother, yet it's okay. I would not bother removing the decal before painting unless it comes off easily.
Though the Dometic cover still looks perfect, our fridge roof vent was also painted but is not holding up as well for the paint is flaking off by the 4 mounting screws. That cover is very smooth plastic, not textured like the Dometic cover which may be the reason.