Sorry I can't help you any more... if you look in web admin in Wizard->Devices->Media Directors, then choose the Computing Application -> Edit button for the MD in question, you will get a dialogue that lists the entries in the Computing menu in Orbiter. One of which is Web Browser and you can see that the script that is run when you choose the browser from LMCE is:
Reading through that script it does a lot of setup for first use and manipulation of profiles and bookmarks, but essentially at the end of the day just runs firefox with no path. I can only assume this means that it is using the first firefox executable in the path when the user is linuxmce (can't see anywhere else an explicit path is being specified, but I stand to be corrected on that).
Doesn't seem very likely, but you could test this by going to KDE desktop, opening a terminal session (which will be the user linuxmce) then just type firefox. If this doesn't play flash properly, then you have found your difference, if it does, then there must be a different issue (which I have no idea on!)