In short, no.
Lavender is safe to use with a babe but you need to think much much lower concentrations than you would yourself - about 0.5%, which amounts to 10 drops EO in 100ml carrier. Obviously you only need about 5 - 10ml of this to completely cover a baby (mind eyes and sensitive bits).
I don´t know about witch hazel and babies, I´d say stay clear as it can be quite astringent. Either put the lavender in water and shake very well before you spray, or put the EO in a carrier oil (remember stay clear of nut oils).
Wether this is strong enough to repel bugs I don´t know, but you shouldn´t use anything stronger on a baby, period.
I think covering up is a better bet.
Sadystar