At what age can you start adding rice cereal to a baby's bottle at night?

Dont need to. No need to add rice to baby's bottle unless it is for a medical reason like reflux. Starting baby foods, that is feeding rice cereal watered down with breast milk or formula and feeding it with a spoon can begin at 4 to 6mo of age.
Why ???????????????? Bottles are for feeding fluids and cereal is used to teach how to eat solids.Rarely we use cereal or extra formula powder to thicken feeds for spitters, but the whole idea of adding solids is to begin the transition to table foods. When baby can sit up and the tongue thrust weakens they are ready for cereals. They also don't sleep better with cereal in their bottles, that is a myth.
ADD CEREAL TO MILK. Adding cereal to milk has problems too.Pls. Be careful as babys can aspirate during coughing. You will know this by auscultation, they have noisy lungs. It is better to spoonfeed the milk/cereal so that baby can swallow it surely.
Rice cereal in milk. You can certainly do it at 4 months of age, which is when the addition of cereal is generally recommended. Some parents have done it as early as 2-3 months without any problems. It's pretty safe. The biggest concern about adding new foods is (1) to make sure the baby's swallow is coordinated so s/he can handle it; and (2) to minimize allergic reactions. Allergies to cereal are rare.