I put kids on bareback or with a plain bellyband to hold onto or any type of saddle, but I'd never strap them in. Kids are remarkably resilient and if your horse is quiet, explain to your daughter that they must both learn about each other. With a saddle or without, but definitely "no attaching of child to horse" let your daughter sit on the horse first. Let her experiment with her own balance up there. Let her slide off into your arms. Let her lean forward, back, side to side, have her do airplane up there... anything at all. Then a few steps forward and a few steps back... once she has found her balance and you have found a comfort zone, you may feel differently.
When I was just a baby, my neighbour had two big ol' Clydes and he would walk them down the road to our house, just so my siblings and I could sit on the horses and be led around the yard. Bareback and hold on to the mane. No one owned a saddle then -- that was for "dem city folks".