The MediaNet browser does not use your company BES at all. You're only using the carrier's network, so you aren't subjected to the web traffic rules that your company has put in place.
The 'Browser' icon is your BES browser... and does filter traffic through your company BES by way of MDS Connection Services on the BES.
If your company didn't want to allow you to use the MediaNet browser, they could very easily disable it. Since they haven't disabled it, then I personally believe that you should go ahead and use it worry-free. (That's my opinion... you may want to verify with your company IT department for a "Real" answer.)