to confirm that you are now running with the updated version.
Then see if the problem still occurs in the same way (no new clues). If it still occurs in the same way, without any new clues that help to solve it then I recommend you look harder at the webserver's access log (right after the problem occurs) to see what was being requested at the time the problem occurred.
You may also want to consider turning on some debugging within ServletExec. This may be done right on the SE Admin UI under Advanced - debug. I recommend turning on the "Request" and the "AS" levels of debugging. Ideally you'd turn this on, make the problem occur, and then turn it off right away. Then go look in ServletExec.log for clues about the problem request(s).
When this problem occurs, are you able to use a new browser instance to request something simple such as /servlet/TestServlet ?
Also, you should consider asking BMC/Remedy (the makers of the web application you are running inside SE) if they know of this issue and it's solution... or if they know of ways to troubleshoot it. It may be possible to "turn on" some level of debugging within the BMC/Remedy application that could help to solve this.