Does anyone know of any sources that say that the reason that we light the Shabbos candles is to show that we do use light for Shabbos, as opposed to the Karaites that do not?
To the best of my knowledge the primary opposition to having pre-lit candles in your home on Sabbath didn't come until the Karaites, hundreds of years after the "Pharisees." (Well the Pharisees had to deal with Zoroastrians who would ban any fire outside of their temples on certain days, but that's an entirely different issue.) I'm inclined to believethat when the folks who brought you the Talmud said the reason for candles was to increase the honor of the day, and make for more harmonious interactions within families (people aren't tripping over each other in the dark), that those are the primary reasons. But it's possible that the mainstream rabbis in the time of the Karaites, i.e. the Gaonim, did further stress the practice to highlight the difference.
I know Chief Rabbi Sacks points out that "Bameh Madlikin" appears in our siddur (prayers) Friday night as a way of emphasizing our belief that flames can stay kindled. (I saw this in his introduction to his British siddur; it's probably in the Koren-Sacks as well.)
I've since heard it observed that the Talmud makes no mention of a blessing on the candles, only that you have to have them lit. The blessing making it clear this is an obligation appears to have been developed by the Gaonim; it's suggested that one rationale in doing so was to stress against Karaite practice.
That's one reason given as to why we eat cholent (which stays on the fire - or nowadays, more likely, in the crockpot - through most of Shabbos). I don't believe I've ever heard of this as a reason for Shabbos candles.