regardless of whether you have the flu jab, you should most definitely have the whooping cough vaccine. we're in the midst of the worst whooping cough epidemic in many, many years and your little one can't be vaccinated against it until they're 2 months old... your vaccination will protect your little one in the interim. whooping cough is terrible for little ones and can be fatal.
don't bother with a balance bike! get a proper bike, remove the chain and cranks, and put the seat down super low. then once your little one is comfortable, you can put the cranks back on and turn it into a proper bike! upcycling etc. Dan