I have done a ton of research on this topic since I have three kids and have long been looking for a Class A with 4 permanent berths suitable for full-timing with my family. Given the research I've done, I'm fairly certain that there are few if any Class A's with double bunks that came that way from the factory. And there aren't a whole lot with one set of bunks either. In general, class A's are mostly geared toward couples, though that's changing a bit.
If you really want a class A and need 5 permanent berths, I think you will have little choice but to find a modifiable floorplan that works for your family and provides room to add the sleeping space you need. I've seen various closets removed to add bunks and even a couple of cases where a couch in a slide was removed to add a set of bunks.
Others mentioned toyhaulers and that is a pretty good option if you are at all handy or can afford a few thousand dollars to hire a carpenter to fit out the space for your specific needs. Toyhauler garages are usually 8-10 feet in length and 8 feet wide, which is certainly big enough to put sleeping for 4 or more.
Someone also mentioned a bus conversion, specifically a design called an "entertainer." These are buses typically used by travelling music groups and feature 6-12 bunks plus lots of seating. Usually there isn't a master bed in the back, but one can be added. There are a lot of these for sale if you google around, but many of these have been ridden hard and put away wet. And these are commercial bus platforms, so be sure you know what you're getting into if you decide to go that route.
Another option, if you're willing to consider it, is a "Super C" class C. Many of these have one set of bunks, plus room for two over the cab, plus a regular bedroom. Most importantly, they have the GVWR to safely carry your family and stuff, unlike a regular class C. Nexus RV just came out with a Super C that includes a U Dinette. I haven't seen one yet,
but here is the floorplan. There are not, unfortunately, a lot of Super C's still in production, but they seem to be making a comeback, or at least are starting too.
Sorry if any of this was stuff you already knew - I err on the side of too much info rather than not enough! Good luck and please post what you find!
