For those who are interested in this thread, I got some help from Marco on this off list, and have increased the number of file desciptors for squid. I had this installed on a developmental (retired, low spec) server, and we used it as the main proxy yesterday afternoon and night. All worked very well.
We updated the main proxy to the new config, made the switch to the updated machine, and we seem to have solved the problem.
I will probably write this up as a howto, but it may have to wait until Christmas Break.