I think another factor (to make it even more complicated) is that you're not working with hard attached axles, but spring hung axles. The flat tire will not only cause one side to sag, but cause the axle springs to react allaround redistributing the weights.
In addition to this, the trailer, weither a 5'r or pull behind, is not supported at four points but five points, the fifth point being the hitch. So, more complexity. Your force drawing needs to be redrawn to show this.
All of this really gets much too complicated to think about (I'm getting a headache just writting about it).
While I too am an engineer (Pulp and Paper Engineering - weired eh), I am also the son of an old time, self taught engineer and farmer. My dad's solution to this problem (if he was still with us) would be to find a DoT or statey officer with four wheel scales, pull the tires on each scale, weigh each point, then flatten one tire and see what happens (also, knowing dad, he would repeat this test at each wheel point).
Maybe not as elegant solution as a mathamatical solution, but sure is a lot easier to do!
Rick & Debbie; Brandy & Schnoodle Dexter & Fritz R.I.P. the Doxie "Kids"
2015 Jayco Pinnacle 36RSQS 5'er
2013 Chevy Silverado 3500HD LTZ, 6.6L Diesel Dually; B&W Goose Neck and Companion 5'er hitch