A week is too short a time to visit the three cities. You will spend so much time traveling from city to city, that you will have little time to truly savor any of them. Also remember you'll be suffering from jet lag. By the time you arrive at a European destination, clear customs and immigration and arrive at your lodgings, half a day is gone and you'll be tired. And usually return flights to the US are early in the morning so that's a travel day, not a tour day.
Your post confuses me. Do you mean the city of Lourdes? The travel time by train from Paris to Lourdes is close to six hours. So you might want to consider staying overnight in Lourdes, especially with a 5 year old child. If you MUST visit another city, I would opt for London because of its proximity to Paris and a 2 hour 20 minute train trip on the Eurostar. If I had only one week and had to visit Lourdes, I would only spend my time in France.
To learn about travel times between cities, you can use the Rome2Rio web site. If you are only traveling Paris to Lourdes, do NOT by a rail pass. If you know your travel dates, buy your tickets on line as far ahead of time as possible to get the cheapest prices. Eurail passes are not accepted on the Eurostar. There is a remarkable amount of information in the Travel Tips section of this website. Take a glance and I'm sure that you can get answers to many of your questions.