My initial thoughts were similar. Were I to do something like this I'd use it as an excuse to play with Mojolicious and its worker queue Minion. I'd make a worker task which would manage the SFTP connection, and you could have your Mojo app implementing the REST API. The worker task would maybe enqueue a keepalive request to itself for whatever interval (and to be fancy if it gets real work before then maybe cancel the prior keepalive and requeue a new one).

</slightly specific handwaving>