Advice for toddler rash around mouth

So my two year old has been suffering with a red rash around his mouth for a number of months, literally came from nowhere.

Our first trip to the doctor, they diagnosed it as impetigo and gave steroid cream, this made no difference at all.

Went back to doctor a couple of weeks later and they diagnosed it as eczema, a different steroid and a moisturiser prescribed. Made no difference.

Third visit to the doctor and she has prescribed a new moisturiser (no steroid) which seems to just be making it angrier.

I feel so sorry for him, nothing the doctor prescribes is making it better and I feel like I keep getting fobbed off.

Can anyone recommend something to help or maybe suggest what it could be? I’m feeling totally helpless.

Thank you 🙏