Flying into City Airport with a bike box can be problematic, the planes are smaller and fitting bulky objects through the doors is by no means a foregone conclusion.
You've just arrived off a transatlantic flight, you have your family, luggage and a bike box? If you can afford it, take a cab, no contest. Your problem will be getting a car big enough but most of the pre-bookable airport transfer services use people carriers, so shouldn't be a problem.
If you REALLY want to go by train, the best route would likely be Heathrow express to Paddington, Paddington to Liverpool Street and then TfL Rail mainline on the Shenfield line to Stratford. No deep platform changes, relatively close, almost certainly the fastest, too.
You might want to tell us which hotel you are staying in - Stratford has relatively few and there some best described as sub-optimal.