It sounds like they are suspecting mastitis then if they are prescribing kesium? Mastitis, you can visually see more how things are going - does it look like she has bad mastitis? It could be that her body is fighting it off (causing the blood results) and if you keep the pups on the affected glands and use hot and cold compresses you could be ok - that is often all that's needed for mild mastitis. But if it's more serious then you'll need to give ABs.
Are the kesium and the amoxicillin separate? Or are they in the same product? (I don't see how they could be the latter, I can't find a product with both.) Personally, I would go to another vet practice and get a second opinion and I wouldn't trust these guys at all. It sounds like they are completely inexperienced dealing with pregnancy and lactation.
The Kesium hasn't been tested as being safe, although there also isn't evidence it is harmful: "Use during Pregnancy, Lactation or Lay
Laboratory studies in rats and mice have not produced any evidence of teratogenic, foetotoxic or maternotoxic effects.
In pregnant and lactating animals, use only according to the benefit/risk assessment by the responsible veterinarian."
https://www.vetuk.co.uk/pet-meds-prescription-only-kesium-chewable-tablets-c-21_1063The amoxicillin is known to be safe and is quite commonly given during pregnancy and lactation, so you could potentially give just that and not the kesium if they are two separate products - but I think you need to consult with a vet who knows what they are doing before changing the meds you've been prescribed.
I would always call a vet out rather than take a lactating bitch to the vet - for infection reasons but also so she can stay with her pups.