All accommodation in Carcassonne is listed on the city's official website. The hotels are grouped by location, with 11 places being listed as 'in or near the mediaeval city'.
The B&Bs aren't grouped by location but some are shown as being in (or near to) the mediaeval city. The first entry, 'La Maison Vieille', has 6 rooms but sleeps 23. Simple arithmetic shows that they must be able to accommodate 4 people together in asingle room. Similarly the third entry, 'Le Grand Puits' (3 rooms, sleeps 11), should also be able to meet your needs. Both of these B&Bs have links to their own websites, which can tell you much more.
Hoping this helps,
Chris