Yep, I had the same issues with stumbling with a small battery.
The alt does help but just look at the wire size that connects the alt to the battery. If your battery is not performing at rated power, the alt wont run the car at optimum because the wire is just not big enough to power everything we stick in our cars.
Think about how much we pull out of our power system. Most have a few gauges, some sort of tuning computer, high flow fuel pump, radio/amps, cell charger, all the dash lights, headlights, tail lights, and the ECU with all the injectors and misc sensors. The AC and ABS pumps pull major power when activated.
If your battery or Alt is not up to peak performance, you will have issues that are intermittent and can be a nightmare to troubleshoot.
Hope you have finally found your problems.