I tried to give DD2 neurofen last night with a syringe - it ended up all over her hair, her clothes and the floor as she didn't want to take it and ended up spitting it everywhere. So, in a nutshell, NO I wouldn't try to give something that I was allergic to to my kids. They are so unpredictable with medicine that you are more than likely going to end up with some of it on you and you will end up having a reaction. Wait the extra time to be on the safe side.
Penicillin is a really common drug suitable for lots of conditions. I wouldn't automatically assume that my child is allergic but I wouldn't give it for the first time without medical supervision. It is worth getting another drug for the moment (to get rid of the chest infection) and then seeing if your doctor would give a controlled small amount of penicillin at a later date to see if your child does have an allergy.
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