There are a number of things that could be happening here and ideally it would be good to have someone there to work through it because while the answers here include things it COULD be, and fit the symptoms, it could be any of them or more and you may end up feeling like you're going round in circles.
Firstly, this behaviour is quite normal even though it's very upsetting. But that doesn't mean it can't be improved. It sounds like there's a good chance that your little girl needs to be near you and held a lot, and therefore a sling would be a great idea. As tiny says, a Moby is a good sling (or any stretchy sling) for keeping her close to you. They take a bit of practise though, and you'll probably need someone to show you how to use it, especially if you want to BF in it. They are BRILLIANT though with a high needs baby. I also find a well fitting pouch to be another useful sling and easier to BF in than a stretchy. I use both with my newborns, the pouch for quick pops in and out and easy BFing and the stretchy sling for carrying all day.
Infant Gaviscon would be the absolute last chance saloon for me. In some babies, the top stomach valve is not matured enough and it causes the baby to reflux - bring up milk and stomach fluids - which can show as the symptoms you describe, together sometimes, with severe vomiting, sometimes non at all. Infant Gaviscon thickens milk and helps to keep it inside the stomach, but, it has its side effects such as constipation and it's also high in salt. Medicating at this stage would not be my first choice. Giving water can cause problems with the gut so constipation is really to be avoided.
The problem is far more likely to be due to your fast milk flow. But, block feeding to reduce your milk supply before this is confirmed could lead you down the route of lowered milk supply so I'd want to work through a few other things first. For example, food intolerance, tongue tie, disorganised suck. Your CS rings bells to me as there's a higher risk of birth trauma. Just being pulled out via CS is more physically traumatic than non-trauma vaginal delivery. Did they try anything else first? Was it a long labour? All these things can contribute to minor birth injuries which take time to heal but can cause great distress in a small baby as well as feeding problems. A trip to a good cranial osteopath may help.
Infacol works by making big bubbles of air in the stomach into small bubbles which are easier to pass. If it's not helping then it's not a wind problem so that's not the solution.
My point is that there are so many possible reasons for these symptoms that while there are some really good ideas here, trying one after another will drive you nuts and if several things are happening at once it's even worse. If the local BFing support people don't know enough to work through these things with you, try a search for La Leche League and see if they have a "leader" nearby, or the NCT, Association of Breastfeeding Mothers (ABM) or Breastfeeding Network (BfN) may have a breastfeeding counsellor nearby who can come to see you. They are all properly qualified.
There WILL be a solution, and I am very happy to talk to you by PM if you want to go into more detail/chat on the phone.