Why only a Red Court Infected? Well, for one thing in my campaigns I don't like my brain exploding from overly-complex metaphysical questions...
My thoughts were you have two demons fighting for control of one (1) host body. Demons, and evil in general, do not work and play well with others...especially other demons. So, it would be a struggle for dominance, and one demon (the stronger) would win out.
Now, there's some possibilties for the character there. Perhaps the winning demon had to cut a deal with the host in order to win decisively? Perhaps the other demon conceded, but suffered less than total destruction? You could even reverse this, and have the character a White Court vampire, but with much greater control over their demon because of a re-negotiated deal with their host.
And by 'skirmishes', I mean gun-toting minions blowing up each others' holdings and possessions and allies. Think of a serious gang war, with bodies everywhere, not just the occasional concession and missing vampire.
If the existence of the character is a 'secret', then the war becomes a little more shadowy...but no less intense. The RCV responsible now knows a powerful secret...one that can lead to a major promotion if they can survive. The White Court wants to silence the RCV permanently, but can't explain why to either the Red Court or their fellow White Court. The RCV could call for aid...but that means giving up their secret and so losing the benefits of the risk taken. And everyone who doesn't know the secret would want to find out why these two groups in this area are in a no-holds-barred battle to the death.