Hi, this isnt a real solution but maybe it helps you:
The Long Beach Challenge Event is available in the Hopper Menu. Every race in this event is a 2 hours race (the driver who drove the most miles in the 2 hours wins). So you have 2 hours or 120 minutes to drive the 125 miles / 201 km. I reached this distance after 80 minutest but you are maybe faster. I think everybody is able to reach the distance in the 2 hours cause in my lobby everybody got it. If you want, you can stop driving after you completed 125 miles / 201 km BUT you have to finish the race to get the Challenge.
The hardest part is to find a race, cause every race take 2 hours. You have to wait a very, very long time until you can drive. I would suggest to watch TV or do something else and go to Forza every 10 minutes to see if the race started.
English isn't my mother tongue, so maybe there are some language mistakes.
Thanks to FF7 Bryan who noticed the Challenge is cumulative and it is possible to drive the 125 miles in more than one race.