Hello and thank you for the question Chris.Answers given are for informational purposes only and are not meant to replace an exam by an in-person.I am Dr. Sohaib, a practicing Internal Medicine Specialist and i will do my best to help you today.
It is extremely unlikely that this rash represent an ongoing H.pylori infection or indicates eradication failure of h pylori.
Infact a rash is not a usual symptom of h.pylori.
In your case the most likely possibility is of a drug induced rash (the antibiotics you took for h pylori eradication).
You should have your treatment reviewed by your doctor, preferably in consultation with a dermatologist and have further management plan chalked out.
May be you need high doses of steroids, and anti-histamines for a longer duration (2 weeks) until this settles down.
The dermatologist may also take a skin biopsy if there are any doubts in the diagnosis; and the biopsy report will clarify the situation to chalk out a more specific and targeted management plan.
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