I should note that I somewhat arbitrarily chose 250 lbs. per side (500 lbs total) to cover our weight up there along with the weight of the camper structure itself, mattress & bedding, and to compensate for the dynamic loading of our moving around.
Formula No. 22:
http://www-classes.usc.edu/engr/ce/457/moment_table.pdf or here as well:
https://archexamacademy.com/download/Structural%20Systems/beam%20diagrams/diagrams%20by%20waterman55.pdf
The variables & constants defined:
https://www.scribd.com/document/226908732/Formulas-3-Beams-and-Diagrams
The two equations of real interest here are the Max Moment and the max deflection (deltamax). Both of these should be algebraically re-arranged to yield a value for I. I'm not sure what to use for max deflection, but I'm tempted to start with .125"
Which ever results in a larger "I" value is the value to use in designing the beam itself. Note that the value for E given in the third link is for steel only. Need to look up the value for whatever the beam is to be constructed of (aluminum, wood, etc.) before proceeding.
FWIW if this sort of thing intrigues you, this is the material covered in a Statics & Strengths of Materials class. If you just want the course book w/o taking the class I'll suggest Cheng's book of the title: "Statics & Strengths of Materials" because it presents the material at a Trigonometry level rather than a Calculus level of math. You'll still likely want to print out one of the first two links and stow it in the book. My class instructor did this for us.
Then need to research how to calculate the "I" of a built-up cross section. So far I'm not finding a simple & concise explanation of that on the net. Cheng's book does cover it.
A note: if you're thinking to build this inner wall structure like a trestle bridge you're making a lot of work for yourself. Keep it simple, build it more like an 'I' beam with box tubing or if wood, square sections for the flanges and a simple sheet or plate for the web.