I believe I figured out the problem. I had to manually install and start the RabbitMQ client.
JBossFuse:admin@root> install mvn:com.rabbitmq/amqp-client/3.3.4
Once that was done, installing and starting the camel-rabbitmq bundle worked.
This is confusing because I don't believe anywhere in the camel-rabbitmq install instructions it says you must first have the RabbitMQ provided client installed.