Hi Summer, I feel you, my baby is 4 months and much the same. It's slightly better now because she is more interactive each week, so it is a bit easier to distract her a bit and you get more positive feedback out of her e.g. more smiles and sometimes the odd laugh too. But still hard.
You say it is colic/ reflux, colic is basically just a term for unexplained crying, but why do you think it is reflux out of interest? If they are showing lots of symptoms of reflux other than the crying I'd go back to GP and be really firm about them prescribing something for reflux for baby. Our DD has 'colic' in the sense that she has unexplained crying but it's no worse in the evenings, Infacol helps a tiny bit but not much, Gripe Water seemed to make it even worse, and I'm confident she doesn't have reflux as she doesn't have any of the symptoms really other than the crying and intermittently being really fussy drinking her bottles. My DD has been checked by GP 3 times and found nothing wrong, and checked for tongue tie at the hospital clinic, and nothing was found. Nobody can find anything wrong with her, yet still she cries for hours every day, often full on red-in-the-face-tears-in-her-ears-screaming-the-place-down. It is HARD.
You say she won't go in the pram, car seat, or sling, do you mean that you have tried all these things and she won't settle at all or that you can't put her in them as she cries? Because DD always cries when we first put her in the pram but after walking for a bit - this can be up to 10/15 minutes if she's really in a state - she will take her dummy, calm down, and go off to sleep. She also sometimes cries when she first goes in the carseat but after driving for a few mins she calms down and goes to sleep, same story with the Baby Bjorn, eventually whilst walking she will calm and go to sleep though she does cry initially. I'm wondering if you have tried this with your DC? Apologies if you have.
I would try different brands of dummy too as someone suggested. DD's dummy is a lifesaver although it doesn't magically solve the problem unfortunately she still cries A LOT.
I had huge problems breastfeeding so by 12 weeks DD was exclusively formula-fed however if you don't have problems I would personally advise to continue the feeding. If you need a break now and then, an occasional bottle of formula could help you but I doubt it will do anything extra for DD. There is no reason to think you have low supply, your DC is putting on weight well and having wet and dirty nappies, therefore they are getting what they need from you, so I don't see how formula could help. A lot of my family put pressure on me to stop BF as if moving to formula would suddenly help DD - it didn't. It made no differnce at all and it is arguably more difficult sometimes in the sense that BFing her would always be a way to stop the crying but bottle feeding is not the same in terms of that comfort and doesn't always work and also you can overfeed so I can't just give her formula every time she cries, and she won't drink if full anyway whereas she would always BF even if only for a few minutes.
Sometimes the hairdryer helps DD. Also might sound silly but she has enjoyed watching children's TV from being about 8 weeks old she would sometimes sit and watch CBeebies or on Youtube she loved the baby sensory videos, worth a try if you haven't already? Have you tried baths also or taking her swimming, some babies find water calming? DD goes swimming once a week now to Water Babies and we were worried initially that she'd scream through it but she doesn't, she sometimes has little bits of whingy crying but not much and generally seems happy and distracted in the water, she also likes being in her Schnuggle bath. Also sometimes the hair dryer noise helps to calm her.
Sorry for the long post, just I know how utterly draining it is and how you feel trapped in the house because you dare not take them anywhere in case they just cry and cry! But you are not alone, there are many of us with babies like this.