Water temperature during the initial steep can make a bigger difference than people think. Cold brew typically uses room temperature or cold water, but if your stomach's still rebelling, try starting with water that's been sitting out or even slightly warm (around 60 - 70°F) for the first few hours, then switch to cold for the rest. This kickstarts the extraction without pulling out as many of the harsh compounds that develop over a really long cold soak.
That said, the other stuff matters too. Lighter roasts actually have more chlorogenic acid than darker roasts, so if you're currently buying light roasts thinking they're "gentler," that's probably backfiring. Switch to a medium or dark roast and see if that helps. You might also want to experiment with your water-to-grounds ratio - going a bit lighter on the coffee concentration means less acid per cup, which is obvious but easy to overlook when you're chasing that rich flavor.
One thing nobody's mentioned: try filtering your final cold brew through a paper filter instead of just pouring straight from the French press. A lot of the micro-particles and oils that remain can irritate your stomach, and a second filter pass catches that stuff. It takes five minutes and might be the cheapest fix before you overhaul your whole method.