I don't think the OP is worried about her baby dying, FreeWorker...frankly (as you were frank....) your anger at 'guilt-inducing rubbish' is misplaced on a thread which seeks practical support for someone who feels conflicted.
OP, it might help you to talk to someone in real life about your options, and help you decide if what you are experiencing is on the normal spectrum for motherhood with a third baby whose siblings still need your care and attention....when it's Christmas and all the extra work that entails....with a baby who is poorly...and where the problems will tend to ease as time goes on. The risk is probably not so much colic/reflux etc with formula, but rather that you will be formula feeding at a time when breastfeeding would have become easier than formula feeding.
There's certainly no way of knowing that a three week old baby switching to formula will be any less needy of your arms, or will sleep better, especially as you say she is not well and does not sleep in her cot (formula may not change that in any way). You would still be feeding often....someone else could feed her of course, if you have someone around who could do that, so you could factor that in.
It's hard - but the only problems I can see that would def be resolved with switching to formula would be that after a couple of weeks you can ditch the breastpads, and you wouldn't smell of milk (most bf women don't smell of milk, IME - but if you don't like it then of course this would disappear, too).
Is there anyone in your life around often enough to help you with your children? Would they be able to step up a bit more?
Hope you can find a way that helps you resolve your dilemma.