If you are willing to switch over to hls (nginx-rtmp supports hls) it'd make your life - from my experience - easier than trying to load balance rtmp itself.
Once you have hls transcoding set up the only thing you need it to either put a cdn in front of your webserver and let that take care of the caching or roll your own using varnish, squid or even nginx yourself (of course there are more possibilities) - HTTP caching is so widespread I'm sure you'll find an easy solution.
If you want to stick with rtmp though, you could set up a similar infrastructure.
Have one master ingest server and multiple edge nodes that each pull from the ingest server.
This setup would be fairly scalable and should work fine for your current load.
Edit: Seems I misunderstood your question: It would probably the easiest to have an api endpoint which you can ask which rtmp server your webcam should stream to instead of trying to load balance.
So once your rtmp server has reached X streams (see nginx-rtmp stat module), you launch a new instance and redirect new streams to that.
nginx-rtmp also has a redirect functionality in on_connect (can't put more than two links yet, just search for on_connect on the directives wiki page) by returning a 3xx header with a Location. I am not sure if this supports redirecting to a different node, but this would be worth a try aswell - can avoid having to manually query before picking a server then.