Thanks again for all of your help. I pursued the network capture route and realized that the other server did not have gzip enabled, which is why it is working. I then did a little searching for "gzip flash player" and bingo, there it is. There is a known problem with IE 6 receiving gzipped data bound for flash. Flash never gets it. IE handles gzipped data correctly in other cases, but there is some sort of communication break down or something. This has been known for going on two years (one blog post was from Feb 2006, another July of this year), so it's not a new problem. There's some debate about it being an IE problem or a Flash problem, but it looks to me like it's an IE problem. Maybe MS decided not to worry about it since IE 7 was on its slow way.
Anyway, I'm glad it's got nothing to do with LSWS!
Now, about disabling gzip in LSWS for a specific url: here is what I believe to be an apache directive for doing it:
SetEnvIf Request_URI "/network\.xml" no-gzip
How would I accomplish the same thing in LSWS?