I'm not really sure how you can argue that a battery saves carbon. I'm assuming you mean carbon emissions. If the battery is charged up from a coal plant, then discharged during the day in lieu of a natural gas turbine it's probably measurably worse in all aspects. Carbon into the atmosphere & numerous other emissions from coal is pretty bad.

Since we're talking about Texas apparently most coal plants by kw-h are either offline or in some cases even decommissioned. Apparently in 2023 13.2% of generation was sourced from coal.