OK here's my first experience with Mountain Lion 10.8.3.

I installed Java 7. Upon trying to install LW (LPE) it gave a message that it needed Java 6 environment and immediately gave me the option to download and install Java 6. I then continued with the LW installation and all went well. I immediately closed LW, installed the security and opened LW and it connected within about 20 seconds.

Do you have Java 6 installed? It seems LW on MacOSX suffers same issue as Windows and some Linux systems in that it will not run directly with Java 7.

Get a hold of Java 6 and see if that makes a difference. The version it installed for me was Java 1.6.0_45 ... since Oracle took over the Java site it has become an incredibly difficult site to navigate and find anything specific on. I found this answer via a google group discussion: ie: some other apps also do not run under Java 1.7.
About Java for OS X 2013-003 or a zipped file download link is Java for OS X 2013-003 (Java 1.6.0_45 update for OSX 10.7 or later.)

Edit: Safe-boot will turn off all but the most necessary system extensions and disable some support files. Safe-boot can be useful for repairing issues or analysing problems to see if there is a conflict of software, etc. Normally you would not install items after a safe-boot.
You will need to change system settings re: default security for OSX 10.8: sample image.
Perhaps the original script was only set to run in 32-bit I don't know. But you could set the installer to run in 32-bit mode via the installer's information window. 32-bit setting sample image