I'm doubting if the small frame (the base) is big enough to avoid the bed from flipping if you stand/sit on the border.
Do not stand on the border!
For adult sitting on edge this looks ok to me, weight of bed base + matress should be sufficient for bed to be stable. But without knowing person's weight it is guesswork.
You will have to build it and test to ...
I can think of a few options:
Screws installed from the back. If you can build the slat panels outside the van and then install them, you can screw through the supports into the back of the slats. This would give you the secure, invisible connection you're looking for.
Nails or staples from the sides. If you're lining the walls of an entire van with slats, ...
Option #1, brads or nails
A brad nailer or pin nailer may be a good solution here, with or without the help of a little adhesive.
The smallest gauges of brads and pins (18-23 gauge) shot from nailers produce very small holes that are considered unobtrusive enough that many don't even bother to fill them. And with 23 gauge the holes are so tiny that they are ...
A scientific calculator or program that gives the Tangent of an angle. Remember, tangent equals opposite length divided by adjacent length. Once you have that ratio you plug in one of the lengths and you have the other length.
I don't know of any DIY truly hidden fastener solutions for this application but there are many proprietary systems for doing so. Just search for "invisible mounting wood slats".
For decks and other larger dimensioned projects you can use a jig to come in at an angle on the inside edge, but I don't see that working out well for this application. ...
…nothing to tell you what dimensions will produce an angle that you supply.
Angles don’t determine the lengths of the sides. You could supply all three angles and that would constrain the sides to a certain ratio, but you’d still need to supply the length of one of the sides to determine the lengths of the other two.
Here is a site for solving triangles.
The user interface doesn't show a right triangle by default (assuming that's what you're after), though the solved triangle does appear to be drawn to scale. Here is what you'll see to start:
Suppose you want a right triangle where, say, you know the bottom side is 96 inches and the angle of elevation is 24 degrees, ...
I created a calculator that searches triangles in 1/8" increments to get as close to the target angle as possible.
As I see it, the advantage of this approach over others is that you can get extremely close to the desired angle while your measurements remain "tape-measure-friendly".
angle = asin(height / hypotenuse)
There is a working demo ...