It may not be 100% what you're looking for and will probably not make your drives silent while streaming but putting L2ARC on that SSD and tweaking prefetch might get you a good way there.

Another much simpler filesystem-agnostic alternative would be to copy it over to the SSD with a script and commence streaming from there. You'll have to wait for the entire file to copy for the stream to start, though. I think some streaming servers may actually support this natively if you mount /var and/or /var/tmp on the faster drive and configure it to utilize it as a "cache".