IMHO, the best, and least used mode, on the camera is Fv mode. It will do exactly what you want. You can choose to manually adjust any 3 of the four (Tv, Av, Exposure, and ISO) and leave the fourth in Auto. In your scenario, with Fv mode, you could set shutter to 1/350, aperture to anything you want, or Auto, dial the exposure comp up or down and the ISO will take care of itself. You can choose to have any, all, or none of the exposure triangle set to manual or auto, and as long as at least one of them is Auto, you can also adjust the exposure compensation.