I am only replying to your post because I am a medical professional and can advise you in that way, but not from a mothering point of view.
First of all, and I don't want to beat you up for this, BUT----NEVER, NEVER, NEVER use medications that have been prescribed for someone else!! Especially if the prescribed medications were for an adult, and now you want to use it on a child. As I totally understand that your intentions were "good", this can be VERY dangerous. The "super-duper" steroid cream that DH used on a *BAD* case of poison ivy is not meant to be used around a child's mouth, especially where there is mucous membrane.
I am glad that you finally did take them to a doctor---however, if something like this is not clearing up by now, it is high time to get them to a dermatologist, who is the expert on funky skin things.
The rash could have more than one single cause---that is, it could have a bacterial component, a fungal component (because it is around her mouth, and the mouth is a mucuous membrane which keeps it more moist than normal skin), an inflammatory component (from her picking at it or rubbing it or licking her lips) and even an allergic component. This is the reason why the antibiotics help some, but don't *cure* it---it does not necessarily mean that the rash is "antibiotic resistant". So, to put them on additional antibiotics seems unnecessary until you find out what the root cause of the problem is. I am not a proponent of prescribing antibiotics willy-nilly if I don't know that the root cause of the problem is bacterial---antibiotics don't work on fungal infections or inflammatory reactions or other causes not related to bacteria, and this is the reason that so many bacteria have become resistant, because they were prescribed for no reason. Because she was picking at the area, that may have caused *some* bacteria to dive into the rash, and that could be why the rash appeared a bit better after the antibiotics. If it was resistant, there would be little to no improvement. I personally happen to think that prescribing anitobiotics without performing a culture and sensitivity on a suspected infection constitutes malpractice---how can you treat something when you don't know what it is?
A dermatologist could do a culture of the areas, which would definitively show what the cause is, and then proper treatment can commence and she'd hopefully get rid of it. I am not going to get involved in your decision to pump, but IMHO, spending the money on a visit to a dermatologist to find the root cause of the rash might be a better investment, since I really don't think that EBM is going to cure the rash, considering everything that you have already been using.
You say that if the rash was all gone you would continue with the Hibiclens wash and antibiotic ointment----why would you continue with antibiotic iontment if the rash is all gone? There is no need to do that. In my opinion, people go WAY overboard on all the antibacterial stuff----you have to be exposed to bacteria and germs to build up your immune system, and using antibacterial stuff and antibiotics all the time is not a good thing to do. The doctor may have recommended Hibiclens because Safeguard is heavy with fragrances which are very irritating by themselves.
Have you tried to just leave it alone, wash it with Ivory soap and let the air get to it, instead of smothering it with antibiotic ointment and steroid cream? It is possible that it could be some sort of localized dermal allergy----when I was a kid, my uncle used to do a lot of fishing and bring my parents fish to cook for dinner. There was one specific kind of fish that would cause me to get a severe rash all around my mouth when I ate it---it would turn beet reed and burn and itch like hell!! And then I would lick it, rub it and scratch it, opening up the skin and turning it into a God-Awful dermal thing. It was not an infection, but it sure looked ugly.
Also---kids around that age lick their lips a lot. If they lick, lick, lick (especially in cold weather), they get a red rash all around their lips that looks pretty ugly. It doesn't mean that it is infected.
Aside from taking the girls to a dermatologist, my recommendation would be to "rule out" the things that could be causing it, since it is a localized reaction, and doesn't seem to be getting worse---just because her sister has it does not mean it is contagious either. They could both have the same allergy or have gotten into the same things that caused the rash to begin with. I am sure you have had close contact with both of them, and if you and DH have not contracted it, it would be safe to assume that it is not contagious. And, I can add to that, STAPH IS VERY CONTAGIOUS!!! If no one else that they have been in contact with has contracted it, I could safely assume that it is not a staph infection. Furthermore, staph infections progress----it is doubtful that a rash that developed over 6 months ago would be in "status quo" mode if it were a bacterial infection. I would start looking at everything they put in their mouth, close to their mouth or that comes in contact with their mouth----food items, liquid drinks, plastic cups, dishwashing detergent, clothing materials, toothpaste, toothbrushes, soap, toys, etc. Also look at outside-of-the-home items too---things in school that they come in contact with, like crayons, pencils, paper, etc. Also---try to find out if any other kids in their school have anything similar. Different types of residue on things can cause skin reactions in people who are sensitive. Do you have any dogs, cats, or other animals? Do you have a rug or carpeting? Sometimes, even things like cleaning product residue on a sink or bathtub is enough to ilicit a reaction in very sensitive people.
Finally---since it is summertime and we can capitalize on nature's healing, maybe you should try to get them out in the sun, without sunscreen on their mouth areas, and let the sun get in the rash for a while to see if that helps. Many types of eczema and psoriasis respond very well to UV light!! You don't have to lay them out in the sun for 6 hours, but enough so that the sun can get to that rashy area. I remember my mother saying that when her kids got diaper rash, the best treatment was to leave the diaper off, lay us on our bellies in the crib and let the sun shine on our butts!! Many times, you are more successful if you let nature treat ailments, especially if you don't know what the ailment is and it is not an immediate threat.
I know that it is difficult, if not damn near impossible, to keep 2 kids that are your kids' ages from putting stuff in their mouths, touching their mouths, licking their lips, etc.---if you had to police that activity, you'd be doing it 24/7. And it probably drives them crazy that you are already washing their mouths like crazy and putting gooey ointment all over it. If it were me, I'd withdraw all "treatment" and start from the beginning. If nothing that you have tried is "curing" it, then that leads me to an allergic cause or some sort of skin condition like eczema. Eczema is a weird thing---it can come on out of the blue, for no good reason whatsoever, and go away on its own or hang around for a few months or a few years. The thing is, sunshine is THE BEST treatment for eczema!! Not hours and hours of skin blistering sun, but a little bit of sun every day. A nice mild soap like Ivory is good---Dove is also very good.
I can hear the frustration in your post----but please remember, your daughters are not dying here, they have a skin rash which is treatable. It is uncomfortable for them, and probably unsightly to look at, but it is not life-threatening or otherwise affecting their internal organs!! All the caregivers of your daughters need to be together on the treatment of this thing---you can't have DH putting steroid cream on it and using Safeguard with all its fragrance in it, and you doing something else, and your mother doing something else. Everybody has to do the same thing to ensure continuity of care.
I hope their rashes go away fast!!