Treating a staph infection on an ornery pet goat (Chlorhexidine)

Our pet goat has white bumps concentrated on her upper front legs. They can be scabby and sometimes have white pus. She doesn't seem bothered by them.
We think it's a staph infection so we’re spraying it 2x a day with Chlorhexidine 2% mixed 50:50 with water.
Our vet is coming in May for vaccinations, so if it's not cleared up by then, she may prescribe something else.
*1 month update*
Vet said to continue with the Chlorhexidine; she didn't feel it was enough to prescribe an antibiotic.
Bumps are greatly improved/reduced. We continue to spot treatment the small amount remaining.
Here’s the Chlorhexidine we’re using:
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Halter:
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Slow feeder treat bowls:
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