I have tried to give you a lot of options here as you are quite late for the event anyway... its the 11th stage today. And all these are very near to where the event starts.
Some three and four stars:
Au Royal Hotel Carcassonne starting at about $70 per night.
Hotel La Bastide again, at about $70.
Hotel Astoria starts at about $90.
Hotel Ibis Carcassonne Centre starts at about $100.
Hotel des Trois Couronnes is a little expensive at about $140 but situated next to the L'Aaude.
Hotel Soleil Terminus Carcassonne situated next to another water body, the canal is at about $125.
Hotel Bristol is also close by at about $100.
If you are ok staying a bit far from the starting point, then there are budget hotels like:
ibis Budget starting at about $75.
Hotel Kyriad is situated on the route of the tour-de-France. Cheap rooms too, starting at about $50.
Hope these help. Have fun.