After watching tshark, poclbm.py output, and tailing the access log for the proxy, I have determined that a low KeepAliveTimeout setting in the Apache config was the culprit.
The KeepAliveTimeout on my server was 3 seconds, and the ask rate was 5. This caused the miner to try to use a KeepAlive session, only to find it timed out. Then it had to establish a new connection. This was giving the connection issues I was seeing.
Essentially what this means is the KeepAliveTimeout setting has to be higher than the ask rate. I would suggest adding a few seconds on top to make sure there's a little wiggle room.
Thanks for the info! I'll add this to the readme, and perhaps to the .htaccess file.
In general, is it preferable to configure Apache with KeepAlive enabled or disabled for this proxy?