You should be able to take that plunket book to your medical centre and ask them to add the vaccines you had as a baby, to your record.
I told my nurse that I had the flu jab at work (which is true) and she just added it to my medical record. She didn't even ask for the exact date or proof of me having had it done. Once you have your vaccines updated online, you can ask them to print you a copy of your records and take it to the immigration doctor.
In regards to vaccines reactions, maybe talk to your doctor and ask for advice. Surely there has been others in your situation. You can get different versions of the same vaccines.