At Versailles in 2008, we rented bikes right on the grounds, at a bike corral at the bottom of the grand stairs in the gardens behind the palace, on the right. As described above, it allowed convenient access to sights (no helmets available, though).
Five years earlier, we rented bikes in Amboise and pedaled out to tour the Chenonceau chateau, really a pleasant ride, after seeing the chateau in Amboise and the town. We were heading to Bordeaux the next day, so didn't see more of the Loire after that.
In Paris, not quite a museum but a worthwhile sight is The Crypt, at the edge of the plaza in front of the Notre Dame Cathedral. Excavated Roman foundations under the modern plaza in this small space include ancient Roman-designed heating and plumbing structures, plus bits of buildings and streets, with modern Paris just overhead. Quite an interesting, but short visit.
We're not 27 anymore, but have done our share of short stays and long-distance stretches between destinations. France is big enough that trips have generally been just to the north, south, east, or west, with the exception of a 9-week trip in 2003. Two months ago, our visit included 3 weeks just in Avignon, then a week a bit down the road based in Saint Remy. As noted earlier, for some folks at least, Provence alone could justify 19 days. And if the Riviera is not of interest (Nice is nice, and the surrounding area was another too-short week of our trip this summer, and we never set foot in a casino or on a beach) I'm curious why you desire going to Saint-Tropez?