It's such a bad idea to do this. Her tummy is already having issues. Cereal has no nutritional value. So every bite of cereal you give your baby is keeping her from getting food. She will be more hungry because she'll be doing without nutrition. She needs her formula or breastmilk.
Her tummy is not able to digest cereal yet so it will sit in her tummy and rot, this will also irritate the reflux. Your doc is not someone I would continue with for this matter. They may not know how reflux in infants is treated.
For reflux you need your doc to write a prescription for a med to relax her tummy so it won't spasm and send the milk back up her esophagus. That is so painful for them.
Once she is doing the meds and takes them a few minutes before each feeding she'll keep her milk down and it will nourish her better. She won't be as hungry.
We had a prescription for Reglan, it smelled like oranges and he took it right down. Then his tummy was relaxed a few minutes later. He could take a bottle and keep every drop down. He stopped being so hungry and started growing better too.
We didn't start him on any cereal until he was a full 6 months old. We made it in a bowl with a teaspoon of rice cereal with about 2-3 table spoons of milk. This made it very runny and he was able to keep it down too.
We also made each and every bottle with Mylanta Gas Drops in it then added the powder formula and swirled it to mix it up. Shaking the bottle adds tons of air and that makes reflux even worse.
So, don't give the baby cereal unless you want to deprive them of nutrition and make life much worse with more puke and more gas.
Formula and breast milk is total nutrition, they need it for their body and brain to develop.
Cereal is filler, it has NO nutrition in it. She needs you to feed her formula or breastmilk.
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As for your ETA we that have kids with reflux know this is not how you treat it successfully. The cereal will make the reflux worse. Much much worse. So a doc who tells you to do this is not very knowledgeable about reflux.
If the problem is a swallowing issue then thicker fluid is a good thing but there are powders that can be added to make a fluid thicker that does not need to digest. It simply goes on through the body and exits.
Small meals a lot more often along with a med that relaxes the tummy is the best treatment for reflux. Not filling the tummy with cereal that is going to sit there and not digest and rot and cause gas and bloating and pain.
Those of us that have kids with reflux know that your plan is going to not help. It will make life worse, we can't just not say anything because you are possibly doing something that is not great for your baby. Maybe even make them hurt.