I made two trips from the West Coast to Europe and the UK last year and I took my bikes because I am difficult to fit (my road bikes are 42 and 44cm).

For overseas travel I use British Airways because they consider bike boxes to be normal luggage and won't charge you hundreds of dollars like the other airlines. The only additional charge is for extra weight, if you decide to pack your box with lots of extra items, like... all your dirty clothes. (On BA or Iberia, you can also get frequent flyer credit for American Airlines)

I've never used that touring company but I was mostly happy I had my road bike. There were a few rainy days I wished I had a cross bike, but I don't think I would have been satisfied with a hybrid. I just swapped my normal tires for Gatorskins and was fine.

Of course, I did all my own wrenching and route planning, so unless you are comfortable doing that, it may be best to stick with what the tour guides recommend.