A few thoughts:
- As much as I love City Airport, the advise is likely to be of limited use for the OP, as there are very few transcontinental flights to City. Moreover, flights tend to be $$$. Gatwick may be indeed better choice than Heathrow, but also much fewer connections. Most transatlantic flights just go to Heathrow.
- As for travelling from Heathrow to Stratford with a boxed bike, there are no good options. It will either be a faff, or pricey, or both. Depending on when you are touching down, a pre-booked cab may be most convenient option. That's true for early morning hours and late at night. Most transcontinental flights land around 6am, I think. Once you are through immigration and got your luggage, peak hour maddness might be in full swing. A nightmare in a car, and not much better on the tube. (With a boxed bike, the tube in the morning peak on a weekday probably is an even bigger nightmare, albeit cheaper)
- I can hear what you're saying about the advantages of staying in Stratford (cheap hotels, olympic pool, tube connection to city). I would still not recommend to stay there. It's a soulless place that doesn't have the London vibe. Tube connections to central London are decent from almost everywhere, and if hotel prices are lower, that's for a reason. Unless your family wants to use the Olympic pool on a daily basis, I would not regard it as key argument in favour of Stratford. As you said, getting there is easy thanks to good transport links...