I chanced upon the answer in a 2005 Flight Global article when I was looking into a hypothesis that the fairing is made of different materials.
The stripe is a flexible joint:
Although the fairing is not theoretically designed to carry fuselage loads, Airbus has built the fairing to absorb the inevitable bending loads that will be transferred from the primary structure of the fuselage to the skin fairing through the belly substructure. The metallic strip visible in the aft section of the fairing on MSN001 is a flexible joint designed to help further absorption of these loads.
My conclusion is that if painted, the paint would crack with flexing, which isn't a good look. Knowing that it's a joint, finding the relevant 2005 Airbus patent was much easier:
An object of the invention is to provide a fairing capable of absorbing compression and bending forces due to the movements of the aircraft and transmitted to the fairing, regardless of the size of said fairing. More specifically, the object of the invention is therefore to reduce the frontal loads received by the fairing. A further object of the invention is to reduce a total weight of an aircraft equipped with a fairing.