Me and the fam (bike riding wife + 5 yr old and 2 yr old) spent last summer in Europe, which included TdF section, since we are both bike racers. We toted our bikes over there ourselves. We hadn’t been to Europe before, and I may have considered a tour company if it was just me and my wife, knowing what I know now I would not hesitate to skip the high cost tour companies and do it myself.
Check out eurocamp or keycamp or any of the british camping companies. the campgrounds in Europe are unbelievably nice, and always at the prime location, plus you get to mingle with lots of Dutch, who love americans and are fantastic people, and very cycling crazy. We camped at Bourg D’Oisans (the town at the bottom of the Alpe D’huez climb) for 2 weeks and from that campground rode Alpe d’huez, Galibier, Les Deux Alpes, Col de Latauret, Col du Glandon, and Col de la Croix de Fer. The point is, you could stay in one alpine town and ride lots of great tour climbs.
If you do the camping thing, the company has the tent (really a big set-up, like 200 sq feet, with 3 separate sleeping areas, a 4 burner gas stove, pots, pans, plates, silverware, fridge, the works), already set up and waiting. All you need to take is clothes, towels and sheets/sleeping bag. Or, if you can rough it even more, just reserve at the campgrounds and take your own stuff.
In the Pyrenees, I cannot recommend enough the B&B we stayed. Owned by a british woman and her dutch boyfriend/common-law-husband, the food was fantastic and the company was great, not to mention the location was good too. I can get you the name if you’re interested.
If it was me, I’d wait until the course was announced and then get on the web and book a place in a campground and follow the tour that way. You’ll meet some great people, which will give you a great cultural experience, and ride some epic climbs. With a rental car, you could even catch a feed zone or two.