You might want to check LogMeIn's website...Where you'll find that they're no help what-so-ever.
The only way to do it is to either bypass the proxy, or to fool it into thinking your traffic is legitimate. The latter is probably right out; you need to connect to LMI, and they're not allowing that. The former one is extremely unlikely unless they've whitelisted some site that will proxy your traffic for you.
Speaking as an admin, I run IP/Traffic reports all the time to watch for viruses and crackers, and just so I know who watches streaming media, so I can mess with them (grey area by our corporate policy, but I can throttle them, or kick them off at certain times of the day, etc, and they can't complain). I would notice a statistical deviation if you (hypothetically) suddenly started tunneling a VPN connection out on port 80, or something.
Setting up the system is challenging, but even a monkey would notice stuff like that once it's up. If you can't convince them to make an exception, I'd give up.