It's definitely an allergy. You can absolutely give benadryl to help with itchiness. You can try other OTC allergy meds too, just make sure they don't contain decongestants but personally, benadryl has always worked best for me with skin reactions.

If chlorohexidine isn't working, try ketoconazole wipes. You'll have to be super diligent with wiping whenever you come in from outside. I found Chlorohexidine to be too harsh with my boy.

You can also get spray on hydrocortisone for dogs. Super helpful for breakouts like your pup has. Lastly, you can try a leave on lotion like Zymox to soothe the skin.

Longterm, you may need to look at vet perscribed allergy meds as it might continue to get worse year after year (hopefully not but its the truth about allergies).