First let me extend you a hale and hearty welcome to the forum.
I believe that many others feel my suggestions for bed frames are underdesigned when it comes to fasteners and wall strength.
Next let me say that you do not want to tie anything solidly across the width of your Sprinter. The body flexes as it rambles down the highway. If you tie anything solid across it will point load the area of the fasteners and may fracture the steel. The Sprinter body steel does seem a bit brittle compared to the older body steels that I've worked with.
That said, presuming you don't plan to sleep in it with multiple adults while bouncing down the road, there is no reason that a bed across the Sprinter cannot be safely supported by the exterior walls and some sheet metal screws. It just isn't that much weight in the grand scheme of things.
If you don't feel that the vertical pieces in the walls provide enough attachment then I would just add a couple down pieces to the wide horizontal support below. Or perhaps bridge the open unsupported area section from horizontal to roof flange as I did with my ladder/misc tiedown re-purposed shelf uprights as shown in the Cheap Tricks. The side support could then be attached to that (those?). There is no way that the weight of a few people will collapse the walls in so supports down to the floor as some people advocate are not necessary.
There are more of my ramblings here.
https://sprinter-source.com/forums/showthread.php?p=223826#post223826
Many people have installed beds in Sprinters. Try searching from the blue bar above. vic
Edit: I recently answered a PM. This information may help clarify what my concerns are about cross pieces being tied solidly.
It really shouldn't take much allowance for the changing crosswise dimension. Across the length of a metal or wood cross piece there should be enough twist and torsional give that it should be fine that way. My concern with the side to side bridging is that it needs to have some allowance for expanding end to end. In my situation for my bed cross bracing I set the ends of my channel into pockets because that was easy for me.
Off the top of my head. If you plan to use a steel or aluminum angle to rest the cross pieces on (similar to a ledger board), then drilling over sized holes for the fasteners should work. My thought would be 1/2" holes for 5/16" fasteners. Slots might even be better in case the movement is greater than I anticipate. I have no data as to the actual potential movement.
To help prevent rattling a couple fender washers sandwiching a rubber washer or spacer could be included. Double locking jam nuts would keep the fastener from loosening. The cushioned washers would keep the structure tightly attached, but would allow it to slide a bit when necessary.
Any design which allows for some end to end movement should be fine. I believe the problems come from shock load when there is no give at all.