I've completed this about 10 times now, and I've managed to reliably finish the escort with anywhere from one tiny sliver left to about 10% health left.
The key seems to be taking on the frigates early. Upgrade yourself to the 80-missile bay, and then let loose on them every chance you get.
You can ignore the early fighters over the frigates; those early fighters do very little damage to the shuttle.
Like the above poster said, when you are getting through the canyon, there is a split second where you can see the incoming fighters; use that chance to unload.
Don't worry about hitting your own shuttle. I've tested shooting at it and it causes no damage to it that I can see.
You want to completely disable at least a couple frigates before you get into the canyon, if you can. Note that while everything else on the frigate is a one-missile-hit, the central shield generators (at least I think that's what they are) will seem invulnerable — that is, they'll flash like shields. Don't give up and unload a couple more missiles into each; you'll find that they do go down like everything else. Perserverance!