Have it load tested, a new battery can be bad too, specially if its a trash manufactured low quality product line like ac delco power etc.
Your car has a partial smart system which shuts off charging to increase fuel mileage. A bad battery shows thse symptoms on such cars, if you load it heavily (accelerate at full power near 5500 rpm, you may also experience misfires)
Also make sure the negative side of circuit is good, shine the lugs and use a thread chaser tap in the holes to clean corrosion or debris, it also can cause issues,
Also have a voltage drop test on both sides done, specially on the alternator case for ground to eliminate that first.