To get a tamed rhyniognatha, you first need to kill a male rhynio and get its pheromone. Then you get 2 high level brontos and have them breed, because the one you select as a host will die. Why brontos specifically? Because the drag weight of the host matters for the rhynio's quality (idk any better synonyms), and brontos are the easiest 100% quality host to tame. Once you have a worthy bronto to sacrifice, you isolate it, put the pheromone in its inventory and lure high level female rhynio to it. I believe you need to get it to 20% health for it to impregnate the bronto.

It will then crave.. like, anything. From hesperornis eggs to gasoline and ELEMENT. If you dont have those, the level will take a big hit. If you dont mind a lil cheating, you can make a backup before having the bronto be impregnated to restart if it craves something absurd early.